fundamental construct for mathematics education john mason

Fundamental Constructsin Mathematics Education

Fundamental Constructs in Mathematics Education is a unique sourcebookcrafted from classic texts, research papers and books in mathematics educa-tion. Linked together by the editors’ narrative, the book provides a fasci-nating examination of, and insight into, key constructs in mathematicseducation and how they link together. The choice of constructs is based on(some of) the many constructs which have proved fruitful in research andwhich have informed choices made by teachers.

This book is divided into two parts: learning and teaching.The first part includes views about how people learn from Plato to Dewey,

as well as constructivism, activity theory and French didactiques. The secondpart includes extracts concerned with initiating, sustaining and bringing to aconclusion learners’ work on mathematical tasks.

Fundamental Constructs in Mathematics Education provides access to awide range of constructs in mathematics education, and will orient thereader towards important original sources. It is a valuable resource foranyone reading literature related to learning mathematics be they lecturer,teacher, adviser or higher degree student.

John Mason is Professor of Mathematics Education at The Open University.Sue Johnston-Wilder is a Senior Lecturer at The Open University.

Companion VolumesThe companion volumes in this series are:

Mathematics Education: exploring the culture of learningEdited by: Barbara Allen and Sue Johnston-Wilder

Researching Your Own Practice: the discipline of noticingAuthor: John MasonAll of these books are part of a course: Researching Mathematics Learning, that is itselfpart of The Open University MA programme and part of the Postgraduate Diploma inMathematics Education programme.

The Open University MA in EducationThe Open University MA in Education is now firmly established as the most popular post-graduate degree for education professionals in Europe, with over 3,500 students regis-tering each year. The MA in Education is designed particularly for those with experience ofteaching, the advisory service, educational administration or allied fields.

Structure of the MAThe MA is a modular degree and students are therefore free to select from a range ofoptions in the programme which best fits in with their interests and professional goals.Specialist lines in management and primary education and lifelong learning are also avail-able. Study in The Open University’s Advanced Diploma can also be counted towards theMA and successful study in the MA programme entitles students to apply for entry into TheOpen University Doctorate in Education programme.

OU Supported Open LearningThe MA in Education programme provides great flexibility. Students study at their own pace,in their own time, anywhere in the European Union. They receive specially prepared studymaterials supported by tutorials, thus offering the chance to work with other students.

The Graduate Diploma in Mathematics EducationThe Graduate Diploma is a new modular diploma designed to meet the needs of graduateswho wish to develop their understanding of teaching and learning mathematics. It is aimedat professionals in education who have an interest in mathematics including primary andsecondary teachers, classroom assistants and parents who are providing home education.The aims of the Graduate Diploma are to:

• develop the mathematical thinking of students;• raise students’ awareness of ways people learn mathematics;• provide experience of different teaching approaches and the learning opportunities

they afford;• develop students’ awareness of, and facility with, ICT in the learning and teaching of

mathematics; and• develop students’ knowledge and understanding of the mathematics which under-

pins school mathematics

How to applyIf you would like to register for one of these programmes, or simply to find out more infor-mation about available courses, please request the Professional Development in Educationprospectus by writing to the Course Reservations Centre, PO Box 724, The Open University,Walton Hall, Milton Keynes, MK7 6ZW, UK or by phoning 0870 900 0304 (from the UK) or+44 870 900 0304 (from outside the UK). Details can also be viewed on our web pagewww.open.ac.uk.

Fundamental Constructsin Mathematics Education

Edited by John Mason andSue Johnston-Wilder

First published in 2004by RoutledgeFalmer11 New Fetter Lane, London EC4P 4EE

Simultaneously published in the USA and Canadaby RoutledgeFalmer29 West 35th Street, New York, NY 10001

RoutledgeFalmer is an imprint of the Taylor & Francis Group

© 2004 Compilation, original and editorial material, The Open University

All rights reserved. No part of this book may be reprinted orreproduced or utilised in any form or by any electronic,mechanical, or other means, now known or hereafterinvented, including photocopying and recording, or in anyinformation storage or retrieval system, without permission inwriting from the publishers.

British Library Cataloguing in Publication DataA catalogue record for this book is available from the British Library

Library of Congress Cataloging in Publication DataFundamental constructs in mathematics education / edited by John Masonand Sue Johnston-Wilder

p. cm.Includes bibliographical references and index.1. Mathematics--Study and teaching (Elementary)–Handbooks, manuals, etc.2. Mathematics--Study and teaching (Secondary)–Handbooks, manuals, etc.I. Mason, John, 1944– II. Johnston-Wilder, Sue

QA11.2.F86 2004510'.7'1--dc22 2003016652

ISBN 0–415–32697–4 (hardback : alk. paper)ISBN 0–415–32698–2 (paperback : alk. paper)

This edition published in the Taylor & Francis e-Library, 2004.

ISBN 0-203-46538-5 Master e-book ISBN

ISBN 0-203-47217-9 (Adobe eReader Format)

Contents ContentsContents

Sources ix

Introduction 1Structure of the book 1What is a construct? 2

SECTION 1

Activating and analysing learning 5

1 Probing thinking 7Introduction 7Early years 7Middle years 16Later years 25

2 Conditions for learning 30Introduction 30Assumptions and theories 30What is learning? 52

3 Analysis of learning for informing teaching 79Introduction 79Theory of didactic situations 79Activity theory 84Constructivisms 92

4 Affect in learning mathematics 99Introduction 99Motivation, intention and desire 99

5 Learners’ powers 115Introduction 115Natural powers 115Discerning similarities and differences 125Mental imagery and imagination 129Generalising and abstracting 132Generalising and specialising 137Conjecturing and convincing 139

6 Learning as action 143Introduction 143Learner action 145Learning phases 161Turning actions into objects 165Learning from examples 173Practice and skills 174

7 Learning what? 181Introduction 181Using powers 181Mathematical thinking 184Mathematical themes 193Mathematical techniques and procedures 196Mathematical topics 198What learners learn 206

SECTION 2

Guiding and directing learning 215

8 Teachers’ roles 217Introduction 217Positioning 217Teachers, learners and mathematics 220Teaching as … 223

9 Initiating mathematical activity 228Introduction 228Principles 228Tasks 238Situations and apparatus 245

vi Contents

10 Sustaining mathematical activity 260Introduction 260Integrating frameworks 260Teacher intervention 266Mathematical discussion 277

11 Concluding mathematical activity 280Introduction 280Reflection 280

12 Having learned … ? 288Introduction 288Knowing 288Understanding 293

Epilogue 311

Bibliography 313Index 331

Contents vii

Sources

The authors and the publishers wish to thank the following for allowing theirwork to be reproduced in the book.

Extracts reprinted by permission of the publisher from Toward A Theory ofInstruction by Jerome S. Bruner, pp. 160, 161, 163, 44-5, 10, 12, 14Cambridge, Mass.: The Belknap Press of Harvard University Press, copyright© 1966 by the President and Fellows of Harvard College.

Hewitt, D., ‘Arbitrary and Necessary, Part 1: A Way of Viewing the Mathe-matics Curriculum’, For the Learning of Mathematics. Extracts reprinted withpermission of the publisher, FLM Publishing Association.

Marton, F. and Booth, S., Learning and Awareness, Erlbaum Mahwah, 1997.Extracts reprinted by permission of the publisher Lawrence ErlbaumAssociates.

Hughes, M., Children and Number: Difficulties in Learning Mathematics,Blackwell, Oxford, 1986. Reprinted by permission of the BlackwellPublishing Limited.

Extracts from The Psychology of Learning Mathematics by Richard R. SkempPenguin Books, 1971, copyright © Richard Skemp, 1971. Reproduced bypermission of Penguin Books Ltd.

Extracts from A Boolean Anthology: Selected Writings of Mary Boole on Math-ematical Education, D. Tahta, Derby, 1972. Reprinted by permission of TheAssociation of Teachers of Mathematics.

Extracts from What We Owe Children: The Subordination of Teaching toLearning by Caleb Gattegno, Routledge & Kegan Paul, London, 1970.Reprinted by permission of Taylor & Francis Books Ltd.

Extracts reprinted with the permission of Scribner, an imprint of Simon &Schuster Adult Publishing Group, and with the permission of CambridgeUniversity Press from The Aims of Education and Other Essays by AlfredNorth Whitehead. Copyright © 1929 by The Macmillan Company; copyrightrenewed © 1957 by Evelyn Whitehead.

Wheeler, D., ‘Teaching for Understanding’, Mathematics Teaching 33, pp.45–7,1965, Association of Teachers of Mathematics, Derby. Extracts reprinted bypermission of The Association of Teachers of Mathematics.

Dewey, J., The Child and the Curriculum & The School and Society, Univer-sity of Chicago Press, 1902. Copyright © J. Dewey, 1902. Extracts reprintedwith permission of the University of Chicago Press.

Dewey, J., How We Think: A Restatement of the Relation of ReflectiveThinking to the Educative Process, © 1933 by D. C. Heath & Company.Adapted with permission of Houghton Mifflin Company.

Teplow, D., ‘Fresh Approaches: Promoting Teaching Excellence’,www.acme-assn.org/almanac/jan97.htm. Extracts reprinted with permissionof the Alliance for CME.

Table from ‘Personal Theories of Teaching’ by D. Fox, Studies in HigherEducation, (1983) 8 (2), reprinted with permission of Taylor & Francis Jour-nals, www.tandf.co.uk.

Figure from Pirie, S. and Kieren, T. (1989), ‘A recursive theory of mathemat-ical understanding’, For the Learning of Mathematics, 9 (3). Reprinted bypermission of FLM Publishing Association.

Two figures taken from ‘A Study of Proof Conceptions in Algebra’ by L.Healey & L. Hoyles in The Journal for Research in Mathematics, July 2000(Figure 1, p. 400 and Figure 5, p. 404) Reprinted with permission from theJournal for Research in Mathematics, copyright © 2000 by the NationalCouncil for Teachers of Mathematics. All rights reserved.

Figure ‘Interconnections’ taken from Thinking Mathematically by Mason,Burton & Stacey © Addison Wesley Publishers Limited 1982, reprinted bypermission of Pearson Education Limited.

Kluwer Academic Publishers for permission to use extracts from thefollowing:

Bell, A. ‘Principles for the Design of Teaching’, Education Studies in Mathe-matics, Kluwer Academic Publishers, 1993.

Christiansen, B. and Walther, G., ‘Task and Activity’ in B. Christiansen, G.Howson and M. Otte, Perspectives in Mathematics Education, Dordrecht,Reidel, 1986 (Kluwer).

Fischbein, E., Intuition in Science and Mathematics: An EducationalApproach, Reidel, Dordrecht 1987 (Kluwer).

Freudenthal, H., Revisiting Mathematics Education: China Lectures, Kluwer,Dordrecht, 1991.

Every attempt was made to contact the copyright holders of third party mate-rial. The publishers apologise for any omissions and will be happy to hearfrom anyone who has been missed.

x Sources

Introduction Introduction

John Mason and Sue Johnston-Wilder

This book is part of a long-term project called Meaning Enquiry in MathematicsEducation whose aim is to draw together in an accessible form the manydistinctions, constructs, and strategies that inform and constitute the practiceof teachers and educators, often without our being aware of it. This bookconcentrates on constructs.

Structure of the book

The first chapter is a collection of classic tasks that have been used repeat-edly in the mathematics education literature. It provides readers with accessto the original (or close to original) description of the task so that they canboth try it out for themselves, and be knowledgeable when they encounterreferences to it in other writings.

The structure of the remainder of the book consists of relatively shortextracts (from a few lines to several paragraphs) from the original authorswho introduced or developed the particular construct, linked together by acommentary. The aim is to make the whole informative, and yet to provideaccess to the original voices, and to create a valuable reference forresearchers and teachers. Sometimes a comment has been inserted into anextract to guide you or to refer you to related extracts. For added interest, wehave taken the unusual step of including the author’s birthplace.

The more we read earlier authors, the more connections we find betweenwhat different people have said in different ways. The result is that it is verydifficult to find a coherent linear path through the vast range of observationsand distinctions. You may wish to read the chapters in this book in order oryou may wish to dip in and out of the book, for example, to follow a partic-ular line of enquiry. To facilitate the latter, at various points in the text wehave included references that point forwards and backwards to relatedtopics set in italics. In addition, separate author and subject indexes areprovided so extracts can be found easily.

Choices

Our interest is in locating and drawing attention to important and informativedistinctions made by authors as a result of their research. In undertaking toextract passages from the lengthy and considered works of authors whohave significantly influenced the development of mathematics education as adisciplined domain of enquiry, choices inevitably have to be made. We weresomewhat surprised to find that it was often remarkably difficult to locate aclear but succinct statement of many of the constructs that we have founduseful and that we wanted to include. There are also many more constructsthat we would have liked to include but space has not permitted.

Recognising a tendency to alight on passages which resonate or challengeour experience, we realise that the passages chosen tell you as much aboutus as they do about mathematics education. Nevertheless, it is our hope thatthis collection will enable colleagues to experience some of the pleasure wehave had in rediscovering the considered thoughts of earlier authors, andthat this will stimulate colleagues to return to some of the original worksfrom which these extracts have been drawn.

What is a construct?

Phenomena

In the context of mathematics education, a phenomenon is something that isdistinguished from the flow of events and recognised as having happenedpreviously: recognising it involves discerning some feature or aspect as (rela-tively) invariant in the midst of other things changing. For example:

• A child is seen using a physical object in order to help her think about aquestion.

• A teacher asks a class a question and pauses, waiting for learners tothink about the question, then invites them to talk with a neighbourabout the question before expecting them to contribute to the whole-class discussion. When one participant suddenly sees a way through andexpresses excitement, others pick up the idea and elaborate on it.

In the first example, you might recognise that ‘using a physical object to helpyou think’ is something that people do naturally, and that teachers couldencourage. In the second example, you might recognise that how long ateacher waits after asking a question is something that could serve as an indi-cator of teacher–learner interaction. Researchers then label the phenomenonin some way for easy reference: in the first case, perhaps use of apparatus oruse of manipulables, and in the second case, wait-time. The label brings thephenomenon into existence as an object of thought, that can be discussedand negotiated between colleagues, and studied by individuals. Researchers

2 Fundamental constructs in mathematics education

then enquire into what is involved in the phenomenon, what makes thephenomenon effective, what effectiveness might mean, what learners expe-rience, links to other aspects of teaching and learning, and so on.

Constructs

Enquiry and study lead to the formulation of constructs both to describe aphenomenon in a recognisable form (thus apparatus and wait-time becomethings) and to account for the phenomenon or to place it in some broader ormore general context. Thus apparatus becomes an enactive mode of interac-tion involving a representation, and wait-time leads to talking-in-pairs as apedagogic strategy to be employed and studied. Class discussion is identifiedas a social phenomenon to study; the mechanism of infectious ideas (a meta-phor for ideas quickly spreading through a class) and a teacher belief or prin-ciple of trying to do for the learners only what they cannot yet do forthemselves are identified. A great deal can come from noticing similarities inseveral observations!

A construct is experienced as an awareness. It is signified by a label for adistinction that has been and can be made. It is an abstraction from experi-ence of a phenomenon. It may be a label for the phenomenon itself, or forsomething that explains or accounts for the phenomenon. It is usually asso-ciated with some action that has physical (a classroom practice), mental(recognition, theorising) and emotional (stimulation, judgement) compo-nents. With use the label comes to be the construct, so it is necessary fromtime to time to experience again the underlying awareness.

Frameworks

Constructs enable practitioners to function. A framework is a label or a set oflabels for a collection of constructs. Labels act as triggers so that, in the midstof a situation, some possibility comes to mind associated with one or moreconstructs. Thus enactive mode of interaction, when combined with mental(iconic) and verbal–symbolic modes, becomes a framework: JeromeBruner’s Enactive–Iconic–Symbolic framework, which can come to informteaching through acting as a reminder that learners need to be weaned fromphysical manipulation of apparatus through the use of mental images andmemories of that enaction, to manipulating symbols which stand for or areexemplified by those images.

Use of frameworks of constructs

Since many constructs are implicit and are below the surface of immediateawareness, bringing them into awareness opens them up for validation,modification, and even replacement. Becoming aware of, adopting, andadapting effective frameworks enables teachers to discern phenomena more

Introduction 3

closely, to distinguish phenomena more sensitively, to notice more than theywould be able to without the framework labels.

Constructs require rich personal experience if they are to be internalisedand if they are to inform practice. Frameworks as collections of labels forconstructs are only useful and effective when the label is founded on richpersonal experience and associated with action. Consequently, it is impor-tant to exemplify the constructs and to base them in significant experience.One way to do this is to begin with a collection of classic research taskswhich have entered the mathematics education literature and been taken upby numerous authors. Readers can then imagine or actually try them out forthemselves, as well as use the constructs to think about the tasks and whatthey found, and employ the research findings to enrich their sense of theconstructs.


Section 1

Activating andanalysing learning

This is the first of the two sections making up the body of the text. It consistsof extracts from a wide range of authors, addressing questions such as ‘Whatdoes it mean to learn mathematics?’ and ‘What is actually learned?’ It isfollowed by a section on guiding and directing learning.

This section begins with a collection of some of the classic tasks used byresearchers to ‘probe’ learners’ understanding. You are encouraged to trythese with the learners with whom you work.

The section continues from Chapter 2 with a variety of constructs used indescriptions of what constitutes learning and of conditions which seem tofoster and sustain learning. The extracts reveal an ongoing struggle to reachsatisfactory definitions of learning, while illustrating a number of differentapproaches and analyses. The consensus is that learners need to be encour-aged and supported in actively taking initiative in their learning. This raisesthe complex matter of motivation and affect, which attracts researchers touse a wide range of subtly different constructs. If learning involves action,then what learners bring to class to enable them to take initiative and toparticipate in those actions are natural powers of sense-making. What islearned is the use and extension of those powers applied to mathematicaltopics. The section ends with constructs which offer a means to exposeunderlying structure in any mathematical topic.

1 Probing thinkingProbing thinking

Introduction

In this chapter we have brought together extracts which include briefdescriptions of research tasks which have been designed to probe thinkingand which have been taken up in the mathematics education literature andreferred to by other workers over the years. The intention of this chapter is togive enough description that the interested reader can try these ‘probes’ withlearners of mathematics.

The probes are arranged broadly in order of age of the learners:

• early years (including lower primary);• middle years;• later years (secondary and tertiary).

However, many of them are accessible to learners across a wide range ofyear groups.

Early years

Children’s invention of written arithmetic

I decided to devise a game in which the children’s written representa-tions would serve a clear communicative purpose. The idea for thisgame arose fairly naturally from my earlier work with boxes and bricks.Young children seemed to be attracted by a closed box containing anumber of bricks, and I thought they might be intrigued by the idea ofputting a written message on the lid of a box to show how many brickswere inside.

The game centred on four identical tobacco tins, containing differentnumbers of bricks: usually there were three, two, one and no bricksinside each tin. After letting the child see inside the tins, I shuffled themaround, and asked the child to pick out ‘the tin with two bricks in’, ‘thetin with no bricks in’ and so on. At this stage the child had no alternative

but to guess. After a few guesses, I interrupted the game with ‘an ideawhich might help’. I attached a piece of paper to the lid of each tin, gavethe child a pen, and suggested that they ‘put something on the paper’ sothat they would know how many bricks were inside. The children dealtwith each tin in turn, its lid being removed so that they could see inside.When they had finished, the tins were shuffled around again, and thechildren were asked once more to identify particular tins and seewhether their representations had ‘helped them play the game’. The Tinsgame thus provided not only a clear rationale for making written repre-sentations, but also an opportunity to discover what children understoodabout what they had done.

I carried out a study in which I played the Tins game with twenty-fivechildren, aged 3 years 1 month to 5 years 10 months. Fifteen of the chil-dren were in the nursery class and ten children in class 1 of a predomi-nantly middle-class school. Each child was seen individually in a smallroom away from the classroom … .

There was little doubt about the popularity of the game. The childrenfound the initial guessing-game intriguing and were excited by the ideaof making representations with paper and pencil. Several of theircomments showed that they were very aware of how this could helpthem, such as: ‘It’s easy now coz I’ve done some writing.’

There was also little doubt that their representations did in fact helpthem play the game. Before they made their representations their abilityto recognise each tin was at chance level, but afterwards their perfor-mance was significantly higher: over two-thirds of the pre-school groupand every child in class 1 was able to identify the tins from theirrepresentations.

(Hughes, 1986, pp. 64–5)

I was impressed by the children’s ability to recognise their representa-tions, and was curious whether they would still be able to recognisethem if some time had elapsed. I therefore returned to the school about aweek later and showed each child the tins bearing the representationsthey had made the previous week. As before, I shuffled the tins, and thechildren had to guess which tin contained which number of bricks.

The results were striking: the children were just as good at recognisingtheir representations a week later as they had been at the time. Thosechildren … who had made idiosyncratic representations and had beenunable to recognise them during the first session, were also unable torecognise them a week later. On the other hand, both Richard and Paulwere still able to recognise their idiosyncratic representations a weeklater, with Richard again spontaneously referring to the ‘tail’ on hisrepresentation of zero.

I also used this second visit to the school to find out whether thosechildren who had initially produced any unrecognisable idiosyncratic


representations would benefit from the chance to have another go.These children – there were seven, all in the pre-school group – werefirst asked: ‘Would you like to try it again?’ Most of the children simplyresponded by saying they couldn’t or wouldn’t think of another way todo it, while those who did try again were no more successful thanbefore. I then suggested the idea of one-to-one correspondence bysaying, for example, ‘Why don’t you make two marks on the tin with twobricks in?’ The response to this was immediate. Five of the sevenadopted the iconic strategy at once from a single example, generalisingwithout further suggestions to the remaining tins. … The other two chil-dren required further examples, but they too eventually adopted therule. All these children were then able to identify their responsescorrectly. Thus, by the end of the study, all twenty-five children hadproduced recognisable sets of responses, with or without prompting.

(ibid., pp. 70–2)

Recognising shapes

For this activity, you need a collection of solid shapes (two copies ofeach). They might be made up from Multilink cubes, or come from a setof prisms and pyramids, they might be packages, etc.

Recognising 3-D shapes: Display one copy of each object, and placethe second copy of one of them in a bag or Feely box [so that everyoneexcept the ‘player’ can see]. Now get someone to feel the object anddescribe what features they are using to identify which of the visibleshapes it is.

This is an example of becoming aware of what you are stressing inorder to identify something.

Recognising 2-D shapes: Make up a collection of shapes, or a pack ofcards with shape drawings … on them, and sort them into groups.Provide a name for each group. Then look at how other people havesorted them, and try to work out the basis of their sorting, providingnames for their groups. Then compare notes.

[ … ]Revealing shapes: Construct a screen so that you can gradually reveal a

large cardboard two-dimensional shape from behind it. Every so often,pause to get learners to discuss all the possible shapes it could be, to makea conjecture, and to say why they think that conjecture might be right.

(Mason, 1990, pp. 9–10)

Probing thinking 9

Covered counting

A collection of objects (bottle tops, cubes, beans, … ) are counted by eachand every learner present, and agreement reached as to the number of them(say 12). Eyes are closed, and some of the objects are hidden under a cloth.The remaining objects are clearly visible.

The task is to figure out how many are hidden when there are sevenvisible and 12 in all. … The child who needs to ‘count all’, and needsperceptual materials, is unable to solve this task as posed. Another childmay be able to count on, saying, ‘seven … eight, nine, ten, eleven,twelve’, while putting up one finger with each number word, beginningat eight. She may then notice that five fingers are up and report five asthe answer. Another child may proceed in almost the same way but mayneed to count the fingers that are up to know that five have beencounted. Yet another child might say, ‘I know that 8 + 4 is 12 so 7 + 5must be 12. So there are five hiding.’ This child is using a thinkingstrategy approach.

(Cobb and Merkel, 1989, quoted in Yackel, 2001, p. 19)

(See Floyd et al., 1982, Block 2 for an equivalent alternative.)

Children making sense

Listening to children, and following their line of thought can be mostrewarding. It can also provide insight into the delicate and lengthy process ofmaking sense through acting according to current conjectures and thenmodifying those conjectures as a result of feedback. In the following extract,a mother and her young child were having lunch when the window cleanerappeared in the garden. The mother went out to the kitchen to get him somewater. While she was there she called out to a neighbour, Pamela.

Child: What did Pamela say?Mother: She’s having to pay everybody else’s bills for the window-

cleaner, ‘cause they’re all out.Child: Why [are] they all out?Mother: ‘Cause they’re working or something.Child: Aren’t they silly!Mother: Well, you have to work to earn money, don’t you?

(Tizard and Hughes, 1984, p. 120)

After an interlude for pudding, the child reverted to the conversation:

3 Child: Umm … she can’t pay everybody’s, er … all the bills tothe window cleaner, can she?


4 Mother: No, she can’t pay everybody’s bills … she sometimespays mine if I’m out.

5 Child: ‘Cause it’s fair.6 Mother: Mm, it is.7 Child: Umm, but where does she leave the money?8 Mother: She doesn’t leave it anywhere, she hands it to the

window cleaner, after he’s finished.9 Child: And then she gives it to us?

10 Mother: No, no, she doesn’t have to pay us.11 Child: Then the window cleaner gives it to us.12 Mother: No, we give the window cleaner money, he does work

for us and we have to give him money.13 Child: Why?14 Mother: Well, because he’s been working for us cleaning our

windows. He doesn’t do it for nothing.15 Child: Why do you have money if you have … if people clean

your windows?16 Mother: Well, the window cleaner needs money doesn’t he?17 Child: Why?18 Mother: To buy clothes for his children and food for them to eat.19 Child: Well, sometimes window cleaners don’t have children.20 Mother: Quite often they do.21 Child: And something on his own to eat, and for curtains?22 Mother: And for paying his gas bills and electricity bill. And for

paying for his petrol for his car. All sorts of things youhave to pay for, you see. You have to earn moneysomehow, and he earns it by cleaning other people’swindows, and big shop windows and things.

23 Child: And then the person who got the money gives it topeople …

It seems until turn 11 the child was under the impression that thewindow-cleaner pays the housewives, and not the other way round. Inthe course of the conversation the relationship between work, moneyand goods is slowly outlined for her, but it is still unclear from her lastremark whether she has really grasped all that has been said.

(ibid., pp. 120–1)

The authors concluded that the conversation of which this extract is part‘reveals something which is characteristic of the slow and gradual way inwhich a child’s understanding of an abstract or complex topic is built up’(ibid., p. 122). It would seem that the child is aware that she has notgrasped the full picture and she returns to the conversation repeatedly. Inthe end the adult closes it down, apparently before the child has fullyresolved it to her satisfaction.

Probing thinking 11

Knowing in action, knowing in words

… in the case of a sling made of a ball attached to a string which thechild whirls around and then throws into a box, it has been found thatthe action is performed successfully at age four to five after several tries,but its description is systematically distorted. The action itself issuccessful: the child releases the ball sideways, its trajectory tangential tothe circumference of the circle described while being whirled. But hemaintains that he released the ball either in front of the box or at thepoint of the circumference nearest to it, or even in front of himself, as ifthe ball pursued a straight line from himself to the box, first passingthrough the diameter of the circle described by his arm.

The reason is first of all that in his eyes the action consists of two sepa-rate actions: whirling, then throwing (and not throwing alone). Second,it is usual to throw a ball into a box in a straight line perpendicular to thebox. What is particularly curious is that although the action can besuccessfully executed at age four to five, a good description of it cannotusually be given before age nine to eleven. The object and the actionitself (hence awareness of the latter) are doubtless perceived but are, asit were, ‘repressed’ because they contradict the child’s preconceivedideas.

(Piaget, 1973, p. 25)

Children invent subtraction

Mark Davies is seven years two months old … . From a very early age hehas displayed clear, mathematical talent and a capacity for abstract,logical thinking.

In order to appreciate the significance of Mark’s discovery, it is neces-sary to recall an occasion in his mathematical development. Once, whenhe was five, he asked me, “What is 1 take away 3?”

Somewhat taken by surprise, I replied, “3 take away 1 is 2, agreed?”He agreed impatiently.“Well,” I continued, “1 take away 3 is minus 2.” (I realise that I may be

reproached for using the term ‘minus 2’ and not the more fashionableterm ‘negative 2’.)

I went on to say that it was rather like climbing up and down on therungs of a ladder, and then attempted, in simple language, an explana-tion, using the ladder application approach, of what amounted to theaddition and subtraction of integers. Mark was fascinated by my answer.

It became apparent over the subsequent months that Mark could workout mentally problems similar to the one that he had posed, though Imade no attempt to consolidate or develop his thinking about negativeintegers.

A few days ago I asked Mark, “What is 16 take away 9?”


He gave the correct answer.“How did you get the answer?” I enquired.“Well,” he replied, “6 take away 9 is minus 3, and 10 take away 3 is 7.”I was intrigued. Immediately I asked, “How would you take 29 from

47?”His reply was, “7 take away 9 is minus 2, 40 take away 20 is 20, and 20

take away 2 is 18.”I asked him whether he could get the answer using a different

method. His reply was, “Yes; 7 take away 9 is minus 2, 40 take away 2 is38, and 38 take away 20 is 18.”

(Davies, 1978, pp. 15–16)

The children were subtracting from 50 using dice and Dienes blocks.They were trying to get to 0. They wrote down 50 and subtracted thenumber shown on the die, using Dienes blocks as a check on theirmental calculation. Jenny had 3 left and shook 5. She said: ‘I can’t takethis away. I would owe two.’ She tried this on a calculator and said: ‘It istake away two.’ She later tried to make other negative numbers, and shecould do this. When given the problem ‘The answer is –1. What is thequestion?’, she produced a pattern of questions:

1 – 22 – 33 – 4……

When asked what needed to be taken away from 100 to give –1, she said:‘Easy … 101.’ She said she always made the second number one bigger.She could use this method when the answer was –2, but not for –3.

(Shuard et al., 1991, p. 16)

Concept learning from instances

Let us set before a subject all of the instances representing the variouscombinations of four attributes, each with three values … an array of 81cards, each varying in shape of figure, the number of figures, color offigure, and number of borders. We explain to the subject what is meant bya conjunctive concept – a set of the cards that share a certain set of attrib-utes values, such as ‘all red cards’, or ‘all cards containing red squares andtwo borders’ – and for practice ask the subjects to show us all the exem-plars of one sample concept. The subject is then told that we have aconcept in mind and that certain cards before him illustrate it, others donot, and that it is his task to determine what this concept is. We willalways begin by showing him a card or instance that is illustrative of the

Probing thinking 13

concept, a positive instance. His task is to choose cards for testing, one ata time, and after each choice we will tell him whether the card is positiveor negative. He may hazard an hypothesis after any choice of a card, buthe may not offer more than one hypothesis after any particular choice. Ifhe does not wish to offer an hypothesis, he need not do so. He is asked toarrive at the concept as efficiently as possible. He may select the cards inany order he chooses. That, in essence, is the experimental procedure.

(Bruner et al., 1956, p. 83)

The authors found four strategies being used in this task: testing one hypoth-esis at a time on each instance; trying to test all hypotheses simultaneouslyon each instance; imagining altering one attribute at a time of a positiveinstance to see if it remains a positive instance; and changing more than oneattribute at once.

Recipes

You need38 cup of flour to make one muffin. You have 35 cups of flour.

If you make all the muffins you can make with this flour, how muchflour will you have left?

(Nunes, webref)

Such questions get low correct responses even from teachers! The lectureson the website includes other examples of probes used.

Conservation and invariance

… take the five-year-old, faced with two equal beakers, each filled to thesame level with water. He will say that they are equal. Now pour thecontents of one of the beakers into another that is taller and thinner andask whether there is the same amount to drink in both. The child willdeny it, pointing out that one of them has more because the water ishigher. This incapacity to recognize invariance of magnitude acrosstransformations in the appearance of things is one of the most strikingaspects of this stage.

(Bruner, 1966, p. 13)

Horizontal and vertical arithmetic

This work began in the mid-1980s when Cobb first conducted a numberof individual interviews with American first and second grade pupilsdesigned to gain an understanding of their concepts of number,including numerical operations. Subsequently, such interviews becamean integral part of our classroom-based research and were conducted byvarious members of the research team.


One question that was posed in a number of interviews is thefollowing: ‘Do you have a way to figure out how much is 16 + 9?’ Theproblem was presented in horizontal format using plastic numerals.Students used a variety of methods, including counting methods. Nomatter which method the students used, virtually all of them answeredwith 25. Later in the same interview the pupils were presented with whatappeared to be a typical school workbook page. One item the pupilswere asked to answer was 16 + 9, this time written in vertical format.

Interestingly, for a number of pupils, the problem written in thisformat was a completely different task from the task they had completedearlier. This time a number of students attempted to use algorithmicprocedures they had been taught in school. While some childrenobtained the correct answer of 25, other children answered with 15, stillothers with 115. What was most disturbing were the responses thesechildren gave when asked about the discrepancy between these answersand their former answers of 25. For example, Cobb (1991) describes anepisode with one girl who answered 15. He pointed out that she hadobtained 25 when solving the problem presented in horizontal format.He had asked her if both answers could be right and if one answer wasbetter. She responded that, if you were counting cookies, 25 would beright but that in school 15 was always right.

(Yackel, 2001, pp. 17–18)

L’âge du capitaine

This task gained such notoriety that it has been used and written about inmany different countries, and generated an entire book on the way learnerstry to make sense of tasks they are given.

Sur un bateau, il y a 26 moutons et 10 chèvres. Quel est l’âge du capitaine?[On a boat there are 26 sheep and 10 goats. What is the age of the captain?]

(Baruk, 1985, p. 23)

All over the world (see, for example, Merseth, 1993, webref) teachers findthat, whereas over the space of several years learners become proficient atadding two-digit numbers, a surprisingly high proportion of the respondersto the captain’s age problem (and others like it) continue to add the numbersin the question. Learners are not used to being asked questions you cannotdo, and if they can see something obvious to do with the numbers, theyoften do it without thinking.

Variants used by Baruk (1985) included:

1 J’ai 4 sucettes dans ma poche droite et 9 caramels dans ma pochegauche. Quel est l’âge de mon papa? [I have 4 lollipops in my rightpocket and 9 toffees in my left. How old is my father?]

Probing thinking 15

2 Dans un bergerie il y a 125 moutons et 10 chiens. Quel est l’âge duberger? [There are 125 sheep and 10 dogs on a farm. How old is theshepherd?]

[ … ]4 Dans une classe, il y a 12 filles et 13 garçons. Quel est l’âge de la

maîtresse? [There are 12 girls and 13 boys in a class. How old is theteacher?]

5 Dans un bateau, il y a 36 moutons. 10 tombent à l’eau. Quel est l’âgedu capitaine? [There are 36 sheep on a boat, but 10 fall in the water.How old is the captain?]

6 Il y a 7 rangées de 4 tables dans la classe. Quel est l’âge de lamaîtresse? [There are 7 rows of 4 tables each in a classroom. How oldis the teacher?]

(Baruk, 1985, p. 25)

There are close similarities with a study by Christine Shiu (1978) in whichlearners with no knowledge of Chinese were given problems presented inChinese and in English (all numbers were in numerals). They scored higheron the Chinese than on the English problems!

Good conjecture, bad data

‘Oh point five times oh point five is oh point twenty-five’‘Oh point four times oh point four is oh point sixteen’‘What is 0.3 × 0.3?’

On this experience, it is a good conjecture that the answer to the first ques-tion must be ‘oh point nine’; however, the conjecture is based on ‘bad data’,for it is unwise to read decimals using ‘teens’.

Middle years

Street arithmetic

Researchers observed and interviewed children selling things on the streetsin Brazil, and contrasted their arithmetical competence on the street withtheir performance in school.

The children were approached by the interviewers on street corners orat markets where they worked alone or with their families. Interviewerschose subjects who seemed to be within the desired age range – school-children or young adolescents – and obtained information about theirage and level of schooling along with information on the prices of theirmerchandise. Test items in this situation were presented in the course of


a normal sales transaction in which the researcher posed as a customer.Purchases were sometimes carried out. In other cases the ‘customer’asked the vendor to perform calculations on possible purchases. At theend of the informal test, the children were asked to take part in a formaltest, which was given on a separate occasion no more than a week later,and by the same interviewer. Subjects answered a total of 99 questionson the formal test and 63 questions on the informal test. Since the itemsin the formal test were based upon questions in the informal test, theorder of testing was fixed for all subjects.

The informal test was carried out in Portuguese in the subject’s naturalworking situation, that is, at street corners or an open market. Testersposed successive questions about potential or actual purchases andobtained verbal responses. Responses were either tape-recorded orwritten down by an observer, along with comments. After obtaining ananswer for the item, testers questioned the subject about his or hermethod for solving the problem.

[ … ]After subjects were interviewed in the natural situation, they were

asked to participate in the formal part of the study, and a second inter-view was scheduled at the same place or at the subject’s house. …

From all the mathematical problems successfully solved by eachsubject (regardless of whether they constituted a test item or not), asample was chosen for inclusion in the subject’s formal test. This samplewas presented in the formal test either as a mathematical operationdictated to the subject (e.g., 105 + 105) or as a word problem (e.g., Marybought x bananas; each banana cost y; how much did she pay alto-gether?). In either case, each subject solved problems employing thesame numbers involved in his or her own informal test. Thus, quantitiesused varied from one subject to another.

[ … ]In order to make the formal test situation more similar to the school

setting, subjects were given paper and pencil and were encouraged touse them. When problems were nonetheless solved without recourse towriting, subjects were asked to write down their answers. Only onesubject refused to do so, claiming that he did not know how to write. …

[ … ]Problems presented in the streets were much more easily solved than

ones presented in school-like fashion. We adjusted all scores to a 10-point scale for purposes of comparability. The overall percentage ofcorrect responses in the informal test was 98.2 (in 63 problems solved bythe 5 children). In the formal-test word problems (which provide somedescriptive context for the subject), the rate of correct responses was73.7%, which should be contrasted with a 36.8% rate of correctresponses for the arithmetic operations.

(Nunes et al., 1993, pp. 18–21)

Probing thinking 17

Sharing

During the course of their work on fractions [the teacher] asked the chil-dren to collect any examples they found at home where fractions wereuseful. And so the ‘interest table’ in the class grew as the childrenproduced their examples. One of them involved sharing two [identical]chocolate biscuits between three children. [Another involved] sharingeight [identical] sausages between five people, and … eight marblesamong five children wanting to play.

(Floyd et al., 1982, pp. 73–4)

‘3 spoonfuls of coffee in a 4-cup machine’ and ‘4 spoonfuls for 6 cups’;which coffee will be stronger?

(Streefland, 1991, p. 50)

When Anja and Monica Fractured come home from school they mayhave an apple each.

But what do you make of the difference in size [in the pictured apples,in which one is large and the other small]?

(ibid., p. 63)

Sorting

Many researchers have used sorting tasks as a means of finding out whatlearners consider to be ‘the same’ and ‘different’, and so revealing whatparticular attributes the learners are attending to.

For example, sort the following statements (each presented on its own card):

Getting several people to sort, and then go round and try to express in wordshow other people have sorted, often opens up fresh ways of seeing.


If 3 boxes of sugar cost £1.95,what does one box cost?

If one box of sugar costs 65 pence,what do 3 boxes cost?

If 3 bags of sugar cost £1.95,what does one bag cost?

If one bag of sugar costs 65 pence,what do 3 bags cost?

If 3 boxes of flour cost £1.95,what does one box cost?

If one box of flour costs 65 pence,what do 3 boxes cost?

If 3 bags of flour cost £1.95,what does one bag cost?

If one bag of flour costs 65 pence,what do 3 bags cost?

Using symbols

Children’s responses (14-year-olds)

(Küchemann, 1981, p. 105)

Algebra perimeter task

Write down an expression for the perimeter of each shape:

Correct responses amongst nearly 1000 14-year-olds are given in the table.

(Küchemann, 1981, p. 102)

The book from which these Küchemann extracts come is Children’s Under-standing of Mathematics:11–16 (Hart et al., 1981). This book was one outcomeof the Concepts in Secondary Mathematics and Science (CSMS) project which isstill the most comprehensive survey of learners’ responses ever made in the UK.

Snowflakes

As a preliminary investigation, nine children of varying abilities aged 11to 15 years were individually interviewed and given the question shownin Figure [1]. Typical answers drawn by children were as follows:

Probing thinking 19

6(i) (Level 1) 11(i) (Level 2) 11(ii) (Level 2) 14 (Level 3)

What can you sayabout aif a + 5 = 8?

What can you sayabout uif u = v + 3 andv = 1?

What can you sayabout mif m = 3n + 1 andn = 4?

What can you sayabout rif r = s + t andr + s + t = 30?

a = 3 92% u = 4 61% m = 13 62% r = 15 35%

r = 30 – s – t 6%

u = 2 14% other values 14% r = 10 21%

94% 68% 64% 38%

3e 4h + t

4h + 1t

2u + 16

u + u + 16

2u + 25 + 16

2n

n2

e

e

eh

t

h h

h 5

6

u u

52

22

Part of thisfigure is notdrawn. Thereare sidesaltogether oflength 2.

n

One per square (Figure 2a): Andrew (11, very able); Maria (13, aboveaverage).

All round the edge (Figure 2b): Dawn (12, low ability).Random distribution (Figure 2c): Wendy (11, very able).

Of the nine, only two produced a random pattern. It was apparent thatmost of them felt compelled to put the snowflakes neatly one per squarebut, possibly, this was influenced by their being asked to draw the distri-bution. Therefore, as a follow-up, the question was changed to amultiple-choice format so that just recognizing the most appropriatedistribution was required.

(Green, 1989, p. 29)

Relative reasoning

Harry is taller than Tom.Harry is smaller than Dick.Who is tallest: Tom, Dick, or Harry?[ … ]Harry is 5 foot tall.Harry is 2 inches taller than Tom.Harry is 6 inches taller than Dick.What are the heights of Dick and Tom?[ … ]


a

X XX X

X

X

XX

X

XXX

X

X X

X

b

X XX

XX

XXXXX

XX

X

XX X

c

XX

XX

X

XX X

XXX

X XX

XX

Figure 2 Pilot test: typical children’s drawings of snowflake distribution

The roof of a small garden shed has 16 squaretiles as in the picture. It begins to snow. After awhile a total of 16 snowflakes have fluttered downonto the roof. Put a cross X for each snowflake toshow where you think they would land on the roof.Explain your answer.

Figure 1 Pilot test: snowflakes drawing question

There are many other possible variables, for instance:

(i) whether it is stated at any point in the problem … that there are threeterms involved … ;

(ii) whether the terms are named in any introductory statement, such as:‘There are three boys called Tom, Dick, and Harry’; and, if so,whether the names appear in an order corresponding to that of thecorrect solution;

(iii) whether the link is actually named twice, or whether it is referred topronominally in respect of the second relation in which it figures –for example, one might say ‘Dick is taller than Harry, who is tallerthan Tom’; or, ‘Dick, who is taller than Harry, is smaller than Tom’.

(Donaldson, 1963, pp. 83–6)

and also

We want to find out the ages of two girls called Jean and May. We knowthat a third girl, Betty, is 15, and that she is 3 years older than one of thetwo girls and 5 years older than the other. If we had one more piece ofinformation we could calculate the ages of Jean and May. What is thatpiece of information?

(ibid., p. 87)

Mr Short and Mr Tall

You can see the height of Mr. Short measured with paper clips. Mr. Shorthas a friend Mr. Tall. When we measure their height with matchsticks:

Mr. Short’s height is four matchsticksMr. Tall’s height is six matchsticksHow many paperclips are needed for Mr. Tall’s height?

The incorrect strategy in which the child concentrated on the differ-ence a – b rather than a/b would result in:

Mr. Short needs two more paperclips than matchsticks so Mr. Tall needstwo more paperclips than matchsticks, so the answer is eight.

Probing thinking 21

The use of this ‘addition’ strategy occurred at the 25–50 per cent level onfour of the most difficult items on the ratio test.

[ … ]… asked to enlarge a rectangle [3 cm by 5 cm] so that the new base

was 12, the height using the addition strategy would be 10 cm. Thisamount makes the figure look like a square, and so produced anobvious distortion. Some children, aware that it was two large, fell backon the method, ‘take two 3 cm pieces and the extra two, answer 8 cm’.

(Hart et al., 1981, pp. 94–5)

Student–teacher ratios

In a certain college there are six times as many students as professors. UsingS for the number of students and P for the number of professors, write downan expression.

… when students incorrectly translate ‘There are six times as manystudents as professors’ as 6S = P, the equation is perceived not assymbolizing a sequence of words (‘six’, [‘times’], ‘students’, ‘professors’)but instead as representing a group of six students associated with oneprofessor. Under this interpretation, the equals sign denotes correspon-dence or association rather than equality, and the letters S and P arelabels for students and professors. With a cognitive science perspective,Davis (1984) attributed the reversal error to a difficulty in selectingbetween two frames: (a) the labels frame, dealing with labels or units(e.g., 1 m = 100 cm); and (b) the numerical-variables equation frame,dealing with relations between numbers (e.g., x = 100y).

(MacGregor and Stacey, 1993, p. 219)

The original study of this problem appears to be in Rosnick and Clement(1980), but it has been widely used and reported on with various explana-tions offered. See MacGregor (1991) for a summary of this and related probesincluding additive ones.

Which angles are bigger?

Each learner is given two copies of the same shape triangle, one small butlarge enough to measure the angles, and one relatively large. Each learner isasked to predict the relative sizes of the angles in the two copies (to exposeconjectures that the angle size depends on the size of the angle arms) and thesum of the angles in each copy. Measurements are made, and the classresults are plotted on a histogram, leading to discussion about the merits ofmeasurement and about the sizes of the angles in the two copies of thetriangle (based on Balacheff (1990), see also p. 83).


Next the learners form teams of three or four. Each team has two copies ofeach of three triangles: one copy is quite small, and one copy is quite large.Again predictions of the angle sum are followed by measurements and theclass results are plotted.

Subsequent discussion is likely to confront individuals with the fact thatthe angle sum appears to be constant (within measurement error) and inde-pendent of triangle size. The discussion may also lead to the conjecture thatthere must be some way to see why the angle sum is always constant.

Warehouse

In a warehouse you obtain 20% discount but you must pay a 15% salestax. Which would you prefer to have calculated first, discount or tax?

(Mason, Burton and Stacey, 1982, p. 1)

The task can be extended: you could change the numbers; or ask whatdifference it makes to the management, or how the government requires thetax to be calculated in a sale.

Proof task

Learners can be asked questions like ‘what convinces you?’ and ‘what do youthink will convince a teacher?’ in the context of both familiar and unfamiliarconjectures. For example, a familiar conjecture to be proved might be

A4: Prove that when you add any 2 odd numbers, your answer is alwayseven. (Write down your answer in the way that would get you the bestmark you can.)

(Healy and Hoyles, 2000, p. 404)

An unfamiliar conjecture to be proved might be

A7: Prove that if p and q are any two odd numbers, (p + q) × (p – q) isalways a multiple of 4. (Write down your answer in the way that wouldget you the best mark you can.)

(ibid., p. 404)

Overleaf is a typical layout for the probes:

Probing thinking 23

A1: Arthur, Bonnie, Ceri, Duncan, Eric and Yvonne were trying to provewhether the following statement is true or false:When you add any 2 even numbers, your answer is always even.

From the above answers, choose the one that would be closest to whatyou would do if you were asked to answer the question.

From the above answers, choose the one to which you think yourteacher would give the best mark.

(ibid., p. 400)

Evens and vowels

You are presented with four cards showing, respectively, ‘A’, ‘D’, ‘4’, ‘7’,and you know from previous experience that every card, of which theseare a subset, has a letter on one side and a number on the other side.You are then given this rule about the four cards in front of you: ‘If a cardhas a vowel on one side, then it has an even number on the other side’.

Next you are told: ‘Your task is to say which of the cards you need to turnover in order to find out whether the rule is true or false [for these cards]’.

(Johnson-Laird and Wason, 1977, p. 143)


Arthur’s answer

a is any whole number

b is any whole number

2a and 2b are any two evennumbers

2a + 2b = 2(a + b)

So Arthur says it is true.

Bonnie’s answer

2 + 2 = 4 4 + 2 = 6

2 + 4 = 6 4 + 4 = 8

2 + 6 = 8 4 + 6 = 10

So Bonnie says it is true.

Ceri’s answer

Even numbers are numbers thatcan be divided by 2. When youadd numbers with a commonfactor, 2 in this case, the answerwill have the same common factor.

So Ceri says it is true.

Duncan’s answer

Even numbers end in 0, 2, 4, 6, or8. When you add any two of these,the answer will still end in 0, 2, 4,6, or 8.

So Duncan says it is true.

Eric’s answer

Let x = any whole number

y = any whole number

x + y = z

z – x = y

z – y = x

z + z – (x + y) = x + y = 2z

So Eric says it is true.

Yvonne’s answer

So Yvonne says it is true.

+

=

Originated by Peter Wason in 1966, this task has been used extensivelywith many variations of context and subjects from school children tomedical doctors. Most find it surprisingly difficult at first. The article goeson to observe that most people say ‘A’ or ‘A and 4’, both of which areincorrect!

Later years

Encountering obstacles

Reasoning is not always convincing enough to overcome deeply heldintuitions:

A group of 17-year-old humanities students were shown, on examples,how to convert periodic decimal expansions of numbers into ordinaryfractions (Sierpinska, 1987).

x = 0.1234123412341234 …

Multiply both sides by 10000: 10000x = 1234.123412341234 …

Subtract the first equality from the second: 9999x = 1234

Divide by 9999: x = 1234/9999

The students were accepting the arguments for expansions like theone above (0.989898 … , 0.121121 … , etc.) but refused to believe that0.999 … = 1 even though it was obtained in an analogous way.

At first the students refused both the reasoning and the conclusion,but later, their attitudes started to differentiate. One student, Ewa, beganto accept the proof as mathematically valid, and the conclusion as math-ematically correct, but refused to accept it as true ‘in reality’.

(Sierpinska, 1994, pp. 78–9)

Birthday

A certain town is served by two hospitals. In the larger hospital about 45babies are born each day, and in the smaller hospital about 15 babies areborn each day. As you know, about 50 percent of all babies are boys.However, the exact percentage varies from day to day. Sometimes it maybe higher than 50 percent, sometimes lower.

For a period of 1 year, each hospital recorded the days on which morethan 60 percent of the babies born were boys. Which hospital do youthink recorded more such days?

Probing thinking 25

• The larger hospital (21)• The smaller hospital (21)• About the same (that is, within 5 percent of each other) (53)

The values in parentheses are the number of undergraduate studentswho chose each answer.

Most subjects judged the probability of obtaining more than 60percent boys to be the same in the small and in the large hospital,presumably because these events are described by the same statistic andare therefore equally representative of the general population. Incontrast, sampling theory entails that the expected number of days onwhich more than 60 percent of the babies are boys is much greater in thesmall hospital than in the large one, because a large sample is less likelyto stray from 50 percent. This fundamental notion of statistics isevidently not part of people’s repertoire of intuitions.

(Tversky and Kahneman, 1982, p. 6)

Knowing and not knowing

I asked the students to solve [the following problem].

In the figure, the circle with center C is tangent to the top andbottom lines at the points P and Q respectively. (a) Prove that PV= QV. (b) Prove that the line segment CV bisects angle PVQ.

The students, working as a group, generated a correct proof. I wrotethe proof on the board [which made use of the figure shown below].

A few minutes later I gave the students [the following problem].

You are given two intersecting straight lines and a point Pmarked on one of them, as in the figure [below] … . Show howto construct, using straightedge and compass, a circle that istangent[ial] to both lines and that has the point P as its point oftangency to the top line.


P

CV

Q

P

CV

Q

P

Students came to the board and made the following conjectures, in order:

(a) Let Q be the point on the bottom line such that QV = PV. The centerof the desired circle is the midpoint of line segment PQ.

(b) Let A be the segment of the arc with vertex V, passing through P,and bounded by the two lines. The center of the described circle isthe midpoint of the arc A.

(c) Let R be the point on the bottom line that intersects the line segmentperpendicular to the top line at P. The center of the desired circle isthe midpoint of line segment PR.

(d) Let L1 be the line segment perpendicular to the top line at P, and L2

the bisector of the angle at V. The center of the desired circle is thepoint of intersection of L1 and L2.

The proof that the students had generated – which both provides theanswer and rules out conjectures a, b, and c – was still on the board.Despite this, they argued for more than ten minutes about whichconstruction was right. The argument was on purely empirical grounds(that is, on the grounds of which construction looked right), and it was notresolved. How could they have this argument, with the proof still on theboard? I believe that this scene could only take place if the students simply

Probing thinking 27

P

V

P

V

Q

P

V

Q

P

R

V

didn’t see the proof problem as being relevant to the constructionproblem.

(Schoenfeld, 1985, pp. 35–6)

Problem solving

Painters are at work, painting and decorating the inner walls of a church.Somewhat above the altar there is a circular window. For a decoration,the painters have been asked to draw two vertical lines at a tangent tothe circle, and of the same height as the circular window; they were then[told] to add half circles above and below, closing the figure. This areabetween the lines and the window is to be covered with gold. For everysquare inch, so and so much gold is needed. How much gold will beneeded to cover this space (given the diameter of the circle); or, what isthe area between the circle and the lines?

(Wertheimer, 1961, p. 266)

Probability

All families of six children in a city were surveyed. In 72 families theexact order of births of boys and girls was G B G B B G.

What is your estimate of the number of families surveyed in which theexact order of births was B G B B B B?

The two birth sequences are about equally likely, but most people willsurely agree that they are not equally representative. The sequence withfive boys and one girl fails to reflect the proportion of boys and girls inthe population. Indeed, 75 of 92 [subjects] judged this sequence to beless likely than the standard sequence (p < .01 by a sign test). Themedian estimate was 30.

(Kahneman and Tversky, 1982, p. 34)

First impressions

A number of years ago, I deliberately put the problem

x dx

x( )2 9−∫as the first problem on a test, to give my students a boost as they beganthe exam. After all, a quick look at the fraction suggests the substitutionu x= −2 9, and this substitution knocks the problem off in just a fewseconds. 178 students took the exam. About half used the right substitu-tion and got off to a good start, as I intended. However, 44 of thestudents, noticing the factorable denominator in the integrand, usedpartial fractions to express x x/ 2 9− in the form A / (x –3) + B / (x + 3).Correct but quite time-consuming. They didn’t do too well on the exam.


And 17 students, noting the ( )u a2 2− form of the denominator, workedthe problem using the substitution x = 3 sin q. This too yields the rightanswer – but it was even more time-consuming, and the students woundup so far behind that they bombed the exam.

Doing well, then, is based on more than ‘knowing the subject matter’;it’s based on knowing which techniques to use and when. If yourstrategy choice isn’t good, you’re in trouble.

(Schoenfeld, 1987, p. 32)

Probing thinking 29

2 Conditions for learningConditions for learning

Introduction

Philosophers, authors, and teachers have tried to describe and define whatthey have understood by learning. That learning is important to us as organ-isms, and particularly as human beings, is not in doubt. But what exactly is it,how can it be recognised, and how can it be fostered and sustained? Theseare difficult questions.

There are marked differences in how teachers and educators fromdifferent cultures approach teaching for learning. For example, MargaretBrown (2001) has suggested there is a tendency in France to want to clarifyand expose underlying theories on which to base the development of prac-tice, and a tendency in America to want everything based logically andexplicitly on research that demonstrates what works and what does not. Incontrast, Anglo-Saxon pragmatic empiricism leads to trying things out andonly later (often much later!) locating, articulating, and clarifying the under-lying theories. An observer can often discern consistencies which seem to actas theories or assumptions about teaching, about learning, about the peoplebeing taught, about the school as institution, and so on.

The purpose of this chapter is to act as reminder that current issues inmathematics education stretch back a long way, and that current practiceshave deep historical roots.

Assumptions and theories

As a teacher, it is easy to dismiss theories as irrelevant to practice, butmany different doers and thinkers have come to the conclusion that everypractice is based on theories, even if these theories are implicit andhidden.

Practice and theory: Leonardo da Vinci

Leonardo da Vinci (1452–1519), famous as artist, scientist, draftsman, engi-neer, inventor and anatomist, wrote that: ‘He who loves practice without

theory is like the sailor who boards ship without a rudder and compass andnever knows where he may cast’ (da Vinci, webref).

Some practices are handed on from generation to generation, as whenyoung children ‘play school’ by standing up and telling their dolls what theyshould know. Still, there are theories underlying these behaviours whichhave deep roots. By probing the depths to locate these deeply embeddedtheories and by trying to make them explicit, it is possible to bring them tothe surface so as to examine them carefully, to challenge and perhaps tomodify them. Otherwise they remain a dominant force inaccessibly hiddenbehind behaviour (see also p. 48).

Observation

Frederick Bartlett (b. Gloucestershire, 1886–1969) pioneered the modernapplication of psychology to educational issues. His principal focus was onthe functioning of memory. He observed that:

… our memories are constantly mingled with our constructions, areperhaps themselves to be treated as constructive in character.

(Bartlett, 1932, p. 16)

… the name, as soon as it is assigned, immediately shapes both what isseen and what is recalled.

(ibid., p. 20)

This is both an echo and a development of a suggestion by Michel deMontaigne (1533–1592) who as a French Seigneur was perhaps the firstperson to publish a large collection of essays based on his observations ofhimself and the world around him: Montaigne said, ‘Human eyes can onlyperceive things in accordance with such Forms as they [already] know’ (deMontaigne, 1588, p. 600).

Another way of saying this is that all observation is actually theory-based,and Norwood Hanson (1924–1967), an English philosopher of science,generalised this to: ‘there is a sense … in which seeing is a ‘theory-laden’undertaking’ (Hanson, 1958, p. 19).

This is mirrored by an observation of Louis Pasteur (1822–1895), theFrench scientist who discovered pasteurisation among many other things:‘Dans les champs de l’observation le hazard ne favorise que les espritspréparés’ (‘Where observation is concerned, chance favours only theprepared mind’) (Pasteur in Oxford Dictionary of Quotations).

Nelson Goodman (b. Massachusetts, 1902–1998) was a philosopher with awide range of interests whose work influenced social scientists and educa-tors concerning the nature of observation. He took Hanson’s idea one stagefurther by suggesting that ‘[facts] are as theory-laden as we hope our theoriesare fact-laden’ (Goodman, 1978, pp. 96–7).

Conditions for learning 31

Kurt Lewin (b. Prussia, 1890–1947) eventually moved to the USA. He isoften referred to as the ‘father of social psychology’. He was a pioneer ofwhat came to be called action research. He is much quoted as saying that‘theory without practice is sterile; practice without theory is blind’ and‘nothing is so practical as a good theory’.

Alfred Orage (b. Yorkshire, 1873–1934) was an influential magazine editor inboth England and New York who prompted the study of psychoanalysis andpsychosynthesis. He wrote ‘The observation of others is coloured by our inabilityto observe ourselves impartially. We can never be impartial about anything untilwe can be impartial about our own organism’ (Orage, 1966, p. 58). This parallelsobservations attributed to Montaigne: ‘When most people speak about them-selves they are not speaking about something they actually know’.

Here then is support for the contention that in order to be sensitive tolearners’ experience, and hence effective in interactions with them, it is vitalto work on sensitising oneself through observation of one’s own experience.‘There is no higher or lower knowledge, but one only, flowing out of experi-mentation’ (attributed to da Vinci). Da Vinci advised: ‘Avoid the teachings ofspeculators whose judgments are not confirmed by experience’ (da Vinci,quoted in Zammatio et al., 1980, p. 133).

This book is intended to inform experimentation. Extracts have beenchosen which are, on the whole, brief and succinct, and therefore alsodense. The challenge for the reader is to look for examples of incidents thatillustrate the constructs and theories presented in the extracts, by examiningpast experience, and also by experimenting in the future. It is to be expectedthat some of these constructs will fit with past experience and inform futurechoices, while others may not do so at first or, indeed, ever.

Human psychology

The term psyche refers to inner experiences and functioning, often equatedwith the mind, but also encompassing emotions and behaviour. Fromancient times people have found it useful to discern structure in inner life,and to seek links and evidence in outer behaviour.

Human psyche: The Upanishads

In the ancient Hindu scriptures, a person is likened to two birds.

Two birdsClose yoked companions,Both share the self-same tree.One eats of the sweet fruit;The other looks on, without eating.

(Rig Veda)


The image of two birds, one eating and the other watching pervades Easternand Western art. There are multiple interpretations (indeed an image isconsidered powerful only when there are several contrasting, even contra-dictory interpretations). The single image serves to hold all of these interpre-tations together. An educational interpretation is that the two birds are formsof attention, and the tree represents the material world of sensation. Theeater is the part of us that gets caught up in doing, while the watcher is aninternal monitor–witness that observes without judging. The watcher hasbeen likened to conscience, which needs to be awakened and developed.The overall purpose of education is to awaken and make use of the secondbird, the ‘inner watcher’. When applied to specific techniques, the notion ofthe inner watcher means having a part of you that is separate from the execu-tion of the technique in order to guide activity, remember goals, and so on.When applied to teaching, it means having a part of you separatelywitnessing the lesson, able to observe and to suggest alternative actions.

In another part of the scriptures, a person is likened to a horse, chariot, anddriver. Later versions change the chariot into a carriage, and then a hansomcab, as appropriate to the times. The owner (originally a charioteer) hires adriver, who has responsibility for maintaining the chariot fabric and tackle,and for looking after the horses. If the chariot body is allowed to decay, if thereins and harness go mouldy or stiff, if the horses become hungry or mangy,or if the driver becomes dissolute, then the chariot cannot be used properly. Ifthe horses are not guided, they will be thrown off course by each suddenmovement, and graze at every opportunity. If the carriage is not maintained, itwill creak and crack under stress. If the driver is not paying attention to wherethe chariot is headed, then the route will be missed, and there may be crashes.


The Katha Upanishad (III, v3–4) offers one reading:

Know thou the atman (self) as the owner of a chariot,The chariot as the body,Know thou the buddhi (intellect) as the chariot driver,And mind as the reins.The senses, they say, are the horses;The objects of sense, what they range over.What then is experience?‘Self, sense and mind conjoined,’ the sage replies.

(adapted from Zaehner, 1966, p. 176)

Reins and harness (mental imagery) enable the driver to guide and direct thehorses (emotions, senses), while the shafts enable the horse’s energy to pullthe chariot. The particular version shown here has five horses for the fivesenses.

The image of the chariot can be useful when preparing lessons and whenteaching by acting as a reminder that behaviour (chariot) is driven byemotion (horses) and guided by intellect (driver by means of the reins andharness (mental imagery).

Complaints about education

Education: Plato

Plato (c.428BC–c.348BC) was an aristocrat. He was a student of Socrates and ateacher of Aristotle (who himself taught Alexander the Great). Plato set downaccounts of Socrates’ dialogues with different students. He also constructedhis own philosophy, which was so influential that it has been said thatwestern philosophy is no more than a series of footnotes to Plato. Here Platopraises the way the Egyptians teach their children arithmetic, implying criti-cism of Greek education.

Athenian: … I maintain that freeborn men should learn … as much as inEgypt is taught to vast numbers of children along with their letters. Tobegin with, lessons have been devised there in cyphering for the veriestchildren which they can learn with a good deal of fun and amusement,problems about the distribution of the fixed total number of apples or


garlands among larger and smaller groups, and the arranging of thesuccessive series of ‘byes’ and ‘pairs’ between boxers and wrestlers as thenature of such contests requires. More than this, the teachers have a gamein which they distribute mixed sets of saucers of gold, silver, copper, andsimilar materials or, in other cases, whole sets of one material.

(Plato, Laws VII, paras 819b to c,in Hamilton and Cairns, 1961, pp. 1388–9)

Failings of current education: Herbert Spencer

It is much easier to see what is wrong about current education than it is to dosomething about it. People have complained about perceived deficienciesfrom earliest recorded discussions of education. Herbert Spencer (b.Derbyshire, 1820–1903) had a lot to say on the subject and he is known asthe ‘father of British sociology’. Lawrence Cremin (b. New York, 1925–1990),himself a distinguished historian of American education, cited Spencer aschief inspiration for people such as John Dewey and Edward Thorndike toconsider learning from the perspective of the learner (see Cremin, 1961).According to Spencer:

Nearly every subject dealt with is arranged in abnormal order: defini-tions and rules and principles being put first, instead of being disclosed,as they are in the order of nature, through the study of cases. And then,pervading the whole, is the vicious system of rote learning – a systemsacrificing the spirit to the letter. See the results. What with perceptionsunnaturally dulled by early thwarting, and a coerced attention to books –what with the mental confusion produced by teaching subjects beforethey can be understood, and in each of them giving generalizationsbefore the facts of which they are generalizations – what with makingthe pupil a mere passive recipient of others’ ideas, and not in the leastleading him to be an active inquirer or self-instructor – and what withtaxing the faculties to excess; there are very few minds that become asefficient as they might be. Examinations being once passed, books arelaid aside; the greater part of what has been acquired, being unorga-nized, soon drops out of recollection; what remains is mostly inert – theart of applying knowledge not having been cultivated; and there is butlittle power either of accurate observation or independent thinking.

(Spencer, 1878, p. 28)

To tell a child this and to show it the other, is not to teach it how toobserve, but to make it the mere recipient of another’s observations: aproceeding which weakens rather than strengthens its powers of self-instruction – which deprives it of the pleasures resulting from successfulactivity – which presents this all-attractive knowledge under the aspect


of formal tuition – and which thus generates that indifference and evendisgust not unfrequently felt … .

(ibid., 1878, p. 79)

After his diatribe against contemporary educational practices, which valuedlanguage (Latin, Greek, English) and arts over science, Spencer argued thatlanguage involves memory (meanings of words), but science (includingmathematics) involves both memory and understanding using reason.

Dangers of rote learning: Augustus de Morgan

Augustus de Morgan (b. India, 1806–1871) became the first professor ofmathematics in the University of London, where he collaborated withGeorge Boole (b. Lincolnshire, 1815–1864) on the development of symboliclogic (see also p. 119). He wrote extensively about the nature of mathematicsand how teaching could be improved.

Mathematics is becoming too much of a machinery; and this is moreespecially the case with reference to the elementary students. They putthe data of the problems into a mill and expect the result to come outready ground at the other end. An operation which bears a close resem-blance to that of putting in hemp seed at one end of a machine andtaking out ruffled shirts ready for use at the other end. This mode isundoubtedly exceedingly effective in producing results, but it iscertainly not soaked in teaching the mind and in exercising thought.

(de Morgan, 1865)

His voice was not alone.

If children fail in other subjects they fail more often here (in arithmetic).It is a subject which seems beyond the comprehension of the rural mind.

In arithmetic, I regret to say worse results than ever before have beenobtained – this is partly attributable, no doubt, to my having so framedmy sums as to require rather more intelligence than before: the failuresare almost invariably traceable to radically imperfect teaching.

(Stafford and Darby, HMI, 1876; quoted in McIntosh, 1977, pp. 92–3;reprinted in Floyd, 1981, pp. 6–11)

Note that these extracts, along with all the extracts in this book from a widerange of authors, could be seen as presenting principles without experiencesfrom which to draw generalisations. However, it is assumed that you, thereader, have a great deal of experience on which to draw; the extracts willeither resonate with, or contradict, either your espoused or your enactedperspective, thus promoting further investigation and probing on your part.


Principles and theories

Recitation as learning: Ignatius Loyola

Ignatius Loyola (b. Basque country, 1491–1556) founded the Jesuits in 1540,though the constitution and detailed structure of the Jesuit teaching methodswere not perfected until after his death. One feature of Jesuit instruction wasoral contact between teacher and student. The teacher initiated lessons witha modified form of lecture, the prelection. First, the general meaning of theentire passage was described; then the meaning and construction of eachclause was thoroughly explained; then, under the term erudition, historical,geographical, archaeological, scientific and mathematical and other informa-tion was provided concerning the passage; this was followed by explanationof the rhetorical and poetical forms and their rules by which the passage wasconstructed; then the Latin in the passage was compared with other passageswritten at other times; finally, moral lessons were drawn.

Each day’s work consisted of a review of the previous day’s passage,followed by in-depth study of the current day’s passage, with opportunitiesto recite to each other and to the whole class, and to discuss with each otherrhetorical, grammatical, historical and scientific points of the lesson (calledconcertations). Each week’s work was reviewed, and the entire year’s workwas also reviewed, with those destined for the Jesuit order undertaking areview by teaching all the material (based on Monroe, 1909, p. 427).

The notion of recitation of lessons remained a cornerstone of educationfor centuries, surviving into the nineteenth century in American universities,where students were called upon to recite portions of the previous day’slecture. Even today the tutorial sessions, which run in parallel with lectures,are called recitations in some countries. The idea that discussion helps tocrystallise learning can trace its roots back at least to the Jesuits, if not toPlato’s academy. Likewise, the view that constant repetition and review ofmaterial enhances learning has ancient roots.

Principles of learning: John Comenius

John Comenius (b. near Prague, 1592–1670) was a highly influential educatorwho championed the use of the vernacular for education. However, his influ-ence was largely indirect, his writings disappearing for two centuries beforesurfacing again in the nineteenth century. He participated in andpropounded the pansophic movement, which attempted to capture andorganise all knowledge. He wanted to give ‘an accurate anatomy of theuniverse, dissecting the veins and limbs of all things in such a way that thereshall be nothing that is not seen, and that each part shall appear in its properplace and without confusion’ (quoted in Monroe, 1909, pp. 483–484).

To this end, Comenius developed a successful method of teaching boththe vernacular and Latin by displaying pictures (the closest a text author


could get to displaying the things themselves) with labels. Each label wasused once within a sentence. This was in reaction to the textbooks of thetime which were entirely in Latin and in which the words were merelytreated as words with rules of grammar and form, not words used to expressideas. In 1641, he was summoned by the English Parliament with a view tosetting up a model school, but the ‘Irish Rebellion’ intervened and nothingcame of it. Although his general methods of teaching (‘according to nature’)did not bear fruit beyond the teaching of language, he put forward a numberof principles, which are recognisable even today, even if you do not agreewith them all.

1 Whatever is to be known must be taught (that is, by presenting theobject or idea directly to the child, not merely through form orsymbol).

2 Whatever is taught should be taught as being of practical applicationin everyday life and of some definite use.

3 Whatever is taught should be taught straightforwardly, and not in acomplicated manner.

4 Whatever is taught must be taught with reference to its true natureand its origin; that is to say, through its causes.

5 If anything is to be learned, its general principles must first beexplained. Its details may then be considered, and not till then.

6 All parts of an object (or subject), even the smallest, without a singleexception, must be learned with reference to their order, their posi-tion, and their connection with one another.

7 All things must be taught in due succession, and not more than onething should be taught at one time.

8 We should not leave any subject until it is thoroughly understood.9 Stress should be laid on the differences which exist between things,

in order that what knowledge of them is acquired may be clear anddistinct.

(Monroe, 1909, pp. 488–9)

Exercising faculties: John Locke

John Locke (b. Somerset, 1632–1704) was a philosopher concerned with thenature of human knowledge and understanding. He was a major forcebehind the notion that education is a discipline, concerned with training thefaculties of the mind rather like muscles are trained through disciplined exer-cise. He and others of his time promoted the view that the process oflearning, rather than the thing being learned, is the important and deter-mining feature of education (based on Monroe, 1909, p. 508).

The business of education is not to make the young person perfect inany one of the sciences, but so to open and dispose their minds as may


best make them capable of any [science], when they shall apply them-selves to it. … [not to gain a] stock of knowledge but a variety andfreedom of thinking; as an increase of the powers and activities of themind, not as an enlargement of its possessions.

[ … ]… mathematics, which therefore I think should be taught all those

who have the time and opportunity, not so much to make them mathe-maticians, as to make them reasonable creatures …

(Locke, 1693; quoted in Monroe, 1909, pp. 518–19)

Locke seemed not to notice that as with any subject, learning can be superfi-cial and rote (see p. 151), or deep and relational (see p. 295). Nevertheless,he introduced a fundamental conundrum of education which several shiftsin discourse and perspective have failed to elucidate in the interveningcenturies, namely, the issue of transfer: how is it that something learned inone situation can come to mind to be used in some fresh situation? Educationproceeds largely by isolating and teaching skills that have proved useful, inthe expectation that learners will then be able to use those skills whereverthey are relevant. If only it were that easy!

Natural development of faculties: Johann Pestalozzi

Some of Herbert Spencer’s ideas (see p. 35 and p. 116) were based on thoseof Johann Pestalozzi (b. Switzerland, 1746–1827), who was one of the first toadvocate education for all, including the poor. According to Spencer,Pestalozzi advocated that:

… education must conform to the natural process of evolution – thatthere is a certain sequence in which the faculties spontaneouslydevelop, and a certain kind of knowledge which each requires during itsdevelopment; and that it is for us to ascertain this sequence, and supplythis knowledge.

(Spencer, 1911, p. 61)

Spencer was not overly impressed with the way that this was carried out inpractice, but he acknowledged that teachers had not fully taken the ideas onboard. Indeed, Jean Piaget’s researches (see p. 92) can be seen in this light aselucidating the natural development of human faculties.

Three laws: Edward Thorndike

Edward Thorndike (b. Massachusetts, 1874–1949) was a behavioural psychol-ogist, and one of the first American researchers to use developments inpsychology to formulate research programmes in education. His view oflearning involved the making of connections between stimulus and response.


From this perspective, he initiated educational psychology as a particularbranch of psychology. He proposed that ‘almost everything in arithmeticshould be taught as a habit that has connections with habits already acquired’(Thorndike, 1922, p. 194), a theme that recurs in other authors (see practicemakes perfect, p. 176).

Thorndike derived three laws of learning from his studies:

1 Exercise and repetition: the more often a skill is repeated, the longerit will be retained.

2 Effect: responses associated with satisfaction are strengthened, whilethose associated with pain are weakened.

3 Readiness: a series of responses can be chained together to satisfysome goal which will result in annoyance if blocked.

(based on Thorndike, 1914)

Thorndike’s perspective arose from and supported a fragmentation ofcontent into components that can be mastered in a relatively short period oftime through training behaviour. The possibility of meaning was ignored onthe grounds that it was not observable as or in behaviour.

Teaching as a cultural activity: James Stigler and James Hiebert

James Stigler and James Hiebert are American educators who played aleading role in the Third International Mathematics and Science Study(TIMSS) of mathematics proficiency in many different countries.

Teaching is a complex system created by the interactions of theteacher, the students, the curriculum, the local setting, and otherfactors that influence what happens in the classroom. The way onecomponent works – say the curriculum – depends on the other compo-nents in the system, such as the teaching methods being used. To saythat teaching is a cultural activity reveals an additional truth: Culturalactivities, such as teaching, do not appear full-blown but rather evolveover long periods of time in ways that are consistent with the stableweb of beliefs and assumptions that are part of the culture. The scriptsfor teaching in each country appear to rest on a relatively small andtacit set of core beliefs about the nature of the subject, how studentslearn, and the role that a teacher should play in the classroom. Thesebeliefs, often implicit, serve to maintain the stability of cultural systemsover time. Just as features of teaching need to be understood in termsof the underlying systems in which they are embedded, so too thesesystems of teaching, because they are cultural, must be understood inrelation to the cultural beliefs rapidly becoming a cultural activity.Children, for example, learn naturally, by hanging around computers.But there still are those for whom learning about computers has the


distinctly non-cultural trait of intentionally and deliberately and self-consciously working through the activity.

(Stigler and Hiebert, 1998, pp. 4–11)

Types of knowledge: Lee Shulman

Lee Shulman (b. Illinois, USA, 1938–) follows in the footsteps of John Dewey,building particularly on the notion of psychologising the subject matter inorder to afford learners access (see p. 45). Here he distinguishes betweencontent knowledge and pedagogical content knowledge, and adds a fewmore distinctions between the different kinds of knowledge necessary toteach effectively, which have been much quoted (and challenged, see forexample, Elbaz, 1983).

A teacher knows something not understood by others, presumably thestudents. The teacher can transform understanding, performance skills,or desired attitudes or values into pedagogical representations andactions. These are ways of talking, showing, enacting, or otherwiserepresenting ideas so that the unknowing can come to know, thosewithout understanding can comprehend and discern, and the unskilledcan become adept. Thus, teaching necessarily begins with a teacher’sunderstanding of what is to be learned and how it is to be taught. Itproceeds through a series of activities during which the students areprovided specific instructions and opportunities for learning, though thelearning itself remains the responsibility of the students. … Although thisis certainly a core conception of teaching, it is also an incompleteconception. Teaching must properly be understood to be more than theenhancement of understanding; but if it is not even that, then questionsregarding performance of its other functions remain moot. …

[To teach effectively the teacher needs:

• teacher knowledge;]• content knowledge;• general pedagogical knowledge, with special reference to those

broad principles and strategies of classroom management and organi-zation that appear to transcend subject matter;

• curriculum knowledge, with particular grasp of the materials andprograms that serve as ‘tools of the trade’ for teachers;

• pedagogical content knowledge, that special amalgam of content andpedagogy that is uniquely the province of teachers, their own specialform of professional understanding;

[ … ]• knowledge of educational contexts, ranging from the workings of the

group or class, the governance and financing of school districts, to thecharacter of communities and cultures;


• knowledge of educational ends, purposes, and values, and theirphilosophical and historical grounds.

(Shulman, 1987, pp. 3–4)

See Shulman (webref) for a short clip of him explaining what he means by‘the scholarship of teaching’.

Metaphors for teaching: Dennis Fox

Often it may seem difficult to speak directly about teaching and learning.Rather, everything is spoken about indirectly, in terms of something else. Forexample, teachers are sheepdogs and learners are skittish sheep, or teachersdeliver skills to waiting learners. When people say ‘teaching is guiding’, forexample, they are using a metaphor, in which one thing (teaching) is said tobe something else (guiding). Sometimes this is given as a simile (teaching islike guiding), or even as an analogy (teaching is like guiding, and learnersare like skittish sheep) in which relationships are carried across from onecontext to another more familiar one. Some metaphors are so deeplyingrained that they are almost impossible to avoid, so habitual have theybecome. These are called frozen metaphors (Lakoff and Johnson, 1980)

Dennis Fox (1983) identified four underlying metaphors – transfer, shaping,travelling and growing – and some of the words that contribute to them.


Transfertheory

Shapingtheory

Travellingtheory

Growingtheory

Verbscommonlyused

Convey,impart,implant,imbue, give,expound,transmit, putover, getacross, tell, …

Develop,mould,demonstrate,produce,instruct,condition,prepare, direct,[ … ]

Lead, guide,direct, help,show, [ … ]

Cultivate,encourage,nurture,develop, foster,enable, bringout, …

Content Commodity tobe transferred,to fill acontainer.

Shaping tools,pattern,blueprint.

Terrain,vantage point,[perspective]

Experiences …

Learner Container tobe filled.

[Matter] (clay,wood, metal)to be shaped.

Explorer,[trouper,prisoner].

[Plant]

Teacher Pumpattendant, foodprocessor,[deliveryagent].

Skilledcraftsman …

Guide … Gardener,[cultivator,pruner,fertiliser]

(Fox, 1983, p. 163)

It can be informative to notice which kinds of words you find yourself usingpredominantly and whether they are consistent with your overall perspective.

Educational beliefs: John Dewey

John Dewey (b. Vermont, USA, 1859–1952) was a pragmatic philosopherand educator who undertook to reform the American education system. Heset up an experimental school attached to the University of Chicago wherehe developed an approach to learning that was based on the experience ofthe learner, and on learners actively making use of their own powers (p. 115)to explore and make sense of the world. The much maligned ‘discoverylearning’ and ‘child-centred education’ were derived from his approach, butbeing overly simplified, soon turned into the opposite of what Deweyintended (see integrated teaching, p. 224). The John Dewey Societycontinues his work, promoting educational practices consonant with hisoriginal ideas and modern versions of them.

Here are brief extracts from the first of five parts of Dewey’s ‘creed’,published in 1897. Its value today lies perhaps most strongly in how we


Transfertheory

Shapingtheory

Travellingtheory

Growingtheory

Standardteachingmethods

Lectures,reading lists,duplicatednotes.

Laboratory,workshop,practicalinstructions,recipes.Exercises withpredictableoutcomes.

Simulations,projects, etc.Exercises withunpredictableoutcomes.Discussions;independentlearning.

Experientialmethods,similar totravellingtheory but lessstructured andmorespontaneous.

Monitoringprogress

Measuringand samplingcontents ofvessel.

Checking sizeand shape ofproduct.

Comparingnotes withtravellingcompanion.

Listening toreflections onpersonaldevelopment.

Explanationof failure –teacher’sview

Leaky vessels,smallcontainer.

[Flawed, faultyraw material]

Blinkeredvision; lack ofstamina.Unadventurous,lethargic.

Poor start;inadequatelyprepared; nowill todevelop.

Explainationof failure –student’sview

Poor transferskills, pooraim.

Poor guides;poorequipment;too manyrestrictionson route.

Poor guides;poorequipment;too manyrestrictionson route.

Restricted diet;unsuitablefood;incompetentgardener.

choose to stress differently in some places, and similarly in others. It is anexcellent but difficult exercise to try to write your own creed.

Article I. What Education IsI believe that all education proceeds by the participation of the indi-vidual in the social consciousness of the race. This process beginsunconsciously almost at birth, and is continually shaping the individual’spowers, saturating his consciousness, forming his habits, training hisideas, and arousing his feelings and emotions. …

I believe that the only true education comes through the stimulation ofthe child’s powers by the demands of the social situations in which hefinds himself. Through these demands he is stimulated to act as amember of a unity, to emerge from his original narrowness of action andfeeling and to conceive of himself from the standpoint of the welfare ofthe group to which he belongs. Through the responses which othersmake to his own activities he comes to know what these mean in socialterms. …

I believe that this educational process has two sides – one psycholog-ical and one sociological; and that neither can be subordinated to theother or neglected without evil results following. Of these two sides, thepsychological is the basis. The child’s own instincts and powers furnishthe material and give the starting point for all education. Save as theefforts of the educator connect with some activity which the child iscarrying on his own initiative independent of the educator, educationbecomes reduced to a pressure from without. It may, indeed, givecertain external results but cannot truly be called educative. Withoutinsight into the psychological structure and activities of the individual,the educative process will, therefore, be haphazard and arbitrary. If itchances to coincide with the child’s activity, it will get a leverage; if itdoes not, it will result in friction, or disintegration, or arrest of the child’snature.

[ … ]I believe that the psychological and social sides are organically related

and that education cannot be regarded as a compromise between thetwo, or a superimposition of one upon the other.

[ … ]… In order to know what a power really is we must know what its

end, use, or function is; and this we cannot know save as we conceive ofthe individual as active in social relationships. But, on the other hand,the only possible adjustment which we can give to the child underexisting conditions, is that which arises through putting him in completepossession of all his powers. … it is impossible to prepare the child forany precise set of conditions. To prepare him for the future life means togive him command of himself; …


In sum, I believe that the individual who is to be educated is a socialindividual and that society is an organic union of individuals. …

(Dewey, webref)

Study: John Dewey

Dewey suggests that ‘the interest in … the successful carrying on of anactivity, should be gradually transferred to the study of objects – their proper-ties, consequences, structures, causes and effects.’ In the next extract, hediscusses the possibilities afforded by education in childhood.

The outcome, the abstract to which education is to proceed, is aninterest in intellectual matters for their own sake, a delight in thinking forthe sake of thinking. It is an old story that acts and processes that at theoutset are incidental to something else develop and maintain anabsorbing value of their own. So it is with thinking and with knowledge;at first incidental to results and adjustments beyond themselves, theyattract more and more attention to themselves till they become ends, notmeans. Children engage, unconstrainedly and continually, in reflectiveinspection and testing for the sake of what they are interested in doing.Habits of thinking thus generated may increase in amount till theybecome of importance on their own account. It is part of the business ofthe teacher to lead students to extricate and to dwell upon the distinc-tively intellectual side of what they do until there develops a sponta-neous interest in ideas and their relations with one another – that is, thegenuine power of abstraction, of rising from engrossment in the presentto the plane of ideas.

(Dewey, 1933, p. 226)

Psychologising the subject matter: John Dewey

Dewey considered that the role of the teacher was not to try to impart know-ledge or instruct directly, but rather to establish and maintain conditions thatsupport and focus learning. He developed the notion of psychologising thesubject matter, by which he meant locating problems and situations thatwould lead learners naturally to confront, resolve, and learn important ideasand techniques, thus underlining his basic approach of starting with theexperiences (past and present) of learners. Dewey invented the notion ofpsychologising the subject matter to mean transforming the subject matterinto a form that matches the learner (see p. 228). He wrote at one point thatthe purpose of education is to organise problem-solving and then teach it tostudents (Dewey, 1933).

One consequence of Dewey’s approach is that teaching a topic is not amatter of sorting it out clearly in your own mind and then expounding it tolearners, but of making contact with learner experience and interest, and


devising new experiences that will prompt learners to confront and resolveimportant and relevant problems.

Much of Dewey’s thinking (and that of most other modern authors) can betraced back to the German philosopher Immanuel Kant (b. Prussia, 1724–1804) who did for modern philosophy what Plato did for ancient philos-ophy: he organised it, delineated problems, considered various stancesand put forward a consistent and reasoned position. This has allowedsubsequent authors to compare and contrast their own positions withthose of Kant. Put another way, Kant said much of what had previouslybeen said, and much of what has since been said, about the way in whichpeople come to know things they did not know before. According toKant, ‘ … all human cognition begins with intuitions, proceeds fromthence to conceptions, and ends with ideas’ (Kant, 1781, p. 429, quoted inPolya, 1962, p. 99). Fischbein and colleagues (see p. 63) have developedthe role of intuition in learning science and mathematics.

It was Kant who also said, though not in so many words, that a successionof perceptions does not necessarily add up to a perception of succession. Inother words, a learner can go through a succession of activities but not endup with any sense of it as anything more than a succession, even a smorgas-bord, of activities. If the second bird (see p. 32) is not awake, then asequence of experiences remains just that, a sequence of experiences.

Rhythms of education: Alfred North Whitehead

Alfred North Whitehead (b. Kent, 1861–1947) was a mathematician andphilosopher, fellow of the Royal Society, and colleague of Bertrand Russell(1872–1970). He was deeply concerned about education and was concernedto locate the unity in various approaches. He is perhaps best known in math-ematics for his attempt, with Bertrand Russell, to place mathematics on firmaxiomatic foundations through deriving the axioms from logic.

Whitehead, in common with many thinkers and researchers, endeavouredto find some structure in the experience of learning. Here he draws attentionto how energy ebbs and flows, and how people go through different phasesin their perspectives on life and hence on learning. He identifies three prin-cipal, interacting phases or rhythms of learning.

Life is essentially periodic. It comprises daily periods, with their alterna-tions of work and play, of activity and of sleep, and seasonal periods,which dictate our terms and our holidays; and also it is composed ofwell-marked yearly periods. These are the gross obvious periods whichno one can overlook. There are also subtler periods of mental growth,with their cyclic recurrences, yet always different as we pass from cycleto cycle, though the subordinate stages are reproduced each cycle. Thatis why I have chosen the term ‘rhythmic’, as meaning essentially theconveyance of difference within a framework of repetition. Lack of


attention to the rhythm and character of mental growth is the mainsource of wooden futility in education. … In relation to intellectual prog-ress [there is] the stage of romance, the stage of precision, and the stageof generalisation.

The stage of romance is the stage of first apprehension. The subject-matter has the vividness of novelty; it holds within itself unexploredconnexions with possibilities half-disclosed by glimpses and half-concealed by the wealth of material. … We are in the presence of imme-diate cognisance of fact, only intermittently subjecting fact to systematicdissection. Romantic emotion is essentially the excitement consequenton the transition from the bare facts to the first realisations of the importof their unexplored relationships. …

In [the stage of precision], width of relationship is subordinated toexactness of formulation. … It proceeds by forcing on the students’acceptance a given way of analysing the facts, bit by bit. New facts areadded, but they are the facts which fit into the analysis.

[ … ]The final stage of generalisation … is a return to romanticism with

added advantage of classified ideas and relevant technique. It is thefruition which has been the goal of the precise training. It is the finalsuccess. …

Education should consist in a continual repetition of such cycles. Eachlesson in its minor way should form an eddy cycle issuing in its ownsubordinate process. Longer periods should issue in definite attain-ments, which then form the starting-grounds for fresh cycles. … Thepupils must be continually enjoying some fruition and starting afresh – ifthe teacher is stimulating in exact proportion to his success in satisfyingthe rhythmic cravings of his pupils.

(Whitehead, 1932, pp. 27–31)

… [I] ask you not to exaggerate into sharpness the distinction betweenthe three stages of a cycle. … Of course, throughout a distinction ofemphasis, of pervasive quality – romance, precision, generalisation, areall present throughout. But there is an alternation of dominance, and it isthis alternation which constitutes the cycles.

(ibid., p. 44)

There are some similarities between Whitehead’s rhythms and the van Hielephases (see p. 63), and the structure of attention (see p. 60). There are alsoresonances with other three-fold frameworks that have been devised toinform teaching preparation and the design and use of tasks (see p. 263).Whitehead also wrote, amongst other things, about the need to develop‘mental habits’ so they become ‘the way in which the mind reacts to theappropriate stimulus … ’ (ibid., p. 42).


Education: Jerome Bruner

Jerome Bruner (b. New York, 1915–) is a psychologist who promulgated thenotion of spiral learning, claiming that you can teach any topic to any personby adapting and pitching the ideas appropriately. He also supported themuch misunderstood and maligned concept of ‘discovery learning’ (see alsointegrated teaching, p. 224). He was one of the people who helped intro-duce Vygotsky’s ideas to the west, particularly the notion of scaffolding(see p. 266), and zone of proximal development (see p. 88). As one of theleaders of the ‘cognitive revolution’ and of cultural psychology, Brunerstudied how people’s personal ‘stories’ influence who they are and howthey act.

In the following passage, Bruner provides a reason for attending to theo-ries, because teaching and learner development are both intimately tied upwith views about how knowledge is gained, created, and communicated, aswell as what knowledge is.

… the heart of the educational process consists of providing aids anddialogues for translating experience into more powerful systems of nota-tion and ordering. And it is for this reason that I think a theory of devel-opment must be linked both to a theory of knowledge and to a theory ofinstruction, or be doomed to triviality.

(Bruner, 1966, p. 21)

Bruner also supports the view that education is about bringing processes tothe surface and expressing them, often in succinct ways, in order to buildon them.

Conditions for learning: Robert Gagné

Robert Gagné (b. Massachusetts, 1916–) is an experimental psychologistwho applies theories to learning. He began by studying military training.Gagné suggested that learning tasks for intellectual skills can be organised ina hierarchy according to complexity: stimulus recognition, response genera-tion, procedure following, use of terminology, discriminations, conceptformation, rule application, and problem solving. The primary significanceof the hierarchy is to identify the prerequisites to facilitate learning at eachlevel. Prerequisites are identified by doing a task analysis of a learning/training task. Learning hierarchies provide a basis for the sequencing ofinstruction.

In addition, Gagné’s theory outlines nine instructional events and corre-sponding cognitive processes:

1 gaining attention (reception);2 informing learners of the objective (expectancy);


3 stimulating recall of prior learning (retrieval);4 presenting the stimulus (selective perception);5 providing learning guidance (semantic encoding);6 eliciting performance (responding);7 providing feedback (reinforcement);8 assessing performance (retrieval);9 enhancing retention and transfer (generalisation).

(Based on Gagné et al., 1992.)

Learning: John Holt

John Holt (b. New York, 1923–1985) was a classroom teacher who paid closeattention to learners and pondered deeply his own observations of childrenin his classrooms. He spearheaded a revival of child-sensitive teaching, butbecame disappointed with the impossibility of transforming ordinaryschools. He turned to the home-schooling movement that he helped found.His many books are as inspiring as they are crystal clear, based on vividaccounts of incidents that altered his awareness. He enquired openly aboutlearners’ experience with a view to improving his own teaching. He was alsoinfluenced by Caleb Gattegno, and by visits to classes in Leicestershire,England, in the 1960s. He was the inspiration for the Education Otherwisemovement of home education. Here he suggests that education which iscompletely teacher and system directed is at best inefficient and at worstdamaging.

… I believe that we learn best when we, not others, are deciding whatwe are going to try to learn, and when, and how, and for what reasonsor purposes; when we, not others, are in the end choosing thepeople, materials, and experiences from which and with which wewill be learning; when we, not others, are judging how easily orquickly or well we are learning, and when we have learned enough;and above all when we feel the wholeness and openness of the worldaround us, and our own freedom and power and competence in it.What then do we do about it? How can we create or help create theseconditions for learning?

(Holt, 1970, p. 95)

As usual, Holt is very demanding in this passage. How can classes of 30, or inmany countries, 50 or more, enable or even permit such learner-centrededucation? Can learners really pick and choose and decide for themselves?Perhaps that is what they are doing anyway, which is why they often do notappear to learn what teachers are trying to teach them.


Learning: Hans Freudenthal

Hans Freudenthal (b. Germany, 1905–1990) spent most of his influentialcareer in the Netherlands. As a mathematician turned educator he workedwith Jean Piaget and with Caleb Gattegno as co-founder of a research societythat still runs biannual conferences (AEIEM). He founded an institute inUtrecht, now known as the Freudenthal Institute, and established theapproach known as realistic mathematics. His weighty tome DidacticalPhenomenology of Mathematical Structures analyses how to psychologise thesubject matter (see p. 45 and p. 230) of mathematics by finding the basis foreach school topic in learners’ experiences of phenomena. He believed thatthe problem should grow out of the situation, and also that the child shouldrecognise the problem in the situation.

Mathematical concepts, structures, and ideas serve to organise phenomena– phenomena from the concrete world as well as from mathematics.

[ … ]Phenomenology of a mathematical concept, a mathematical structure,

or a mathematical idea means, in my terminology, describing [a]nooumenon [thought object] in relation to the phainomena [originalLatin spelling] of which it is the means of organising, indicating whichphenomena it is created to organise, and to which it can be extended,how it acts upon these phenomena as a means of organising, and withwhat power over these phenomena it endows us. If in this relation ofnooumenon and phainomenon I stress the didactical element, that is, if Ipay attention to how the relation is acquired in the learning–teachingprocess, I speak of didactical phenomenology of this nooumenon.

(Freudenthal, 1983, pp. 28–9)

Freudenthal prefers to see the development of concepts as the constitutionof mental objects through experience of phenomena rather than throughconcrete embodiments (see p. 203 and p. 249).

Philosophical perspectives: Philip Ballard

Different philosophical perspectives or stances might be thought to be luxu-ries decorating the surface of education, but, in fact, educational principlesare considered to drive behaviour, even when those principles are hiddenbeneath the surface. For example, Philip Ballard (b. Wales, 1865–1950) wasthe author of many textbooks in the 1920s on mathematics education, espe-cially concerning early years, which were popular enough to go into severaleditions. Here Ballard comments on the very different approaches toteaching arithmetic to be found in England and the United States in the1920s, but he shows that the roots of these approaches go very deep intoGreek schools of philosophy:


[England:] … follow tradition in stressing the rational side of arithmetic.After making a few concessions to the immature mind, which may beallowed to gain its first notions of number from bricks or beans, theyhasten to apply general principles to particular instances. They proceeddeductively. Every step at every stage has to be reasoned out. No unex-plained process is allowed to be taken on trust and used on the soleground that the process works. No lumps of knowledge, however usefulthey may be as they are, are allowed, even for a while, to escape thegrinding of the logical machine. And arithmetic is almost made to appearas a mere branch of deductive logic.

[America:] … arithmetic is presented as an inductive science. Reasoningstarts with the concrete fact and ends with the concrete fact. Childrenlearn arithmetic by working sums. The justification for the mode ofprocedure is that the answer is right. The ground for believing theanswer to be right is the word of the teacher, or the result got byreversing the process, or, in the last resort, the irrefutable evidenceafforded by counting. The child does a thing first and understands itafter. Doing is the important thing; and practice in doing – the practicethat, ‘line upon line, here a little and there a little’, fixes deeper anddeeper a series of habits. Arithmetic is in fact not so much an applicationof broad general principles as an organisation of habits.

Thus we have on the one hand the English view that arithmetic islogic, and the American view that it is habit. The contrast is interestingand significant; but it is not new. It resembles, in fact, the antithesisbetween the Platonic and the Aristotelian views of virtue. To Plato virtueis knowledge; to Aristotle it is habit. To Plato it is an intellectual grasp ofthe consequences of our acts; to Aristotle it is the practice of choosingthe mean between two extremes.

These differences in emphasis and outlook are not of mere theoreticalimport: they vitally affect practice. They prescribe what we shall teach,how we shall teach it, and how we shall test it.

(Ballard, 1928, pp. xi–xii)

The American approach developed over some 100 years as American arith-metic textbook authors imported ideas based on Pestalozzi (see below) anddeveloped what they first called the inductive system of teaching, laterrenamed as synthetic (in contrast to analytic or deductive). Of course withina generation even newer textbooks were advocating a fusion of the analyticand synthetic methods (Nietz, 1966).

For a comprehensive selection of extended extracts from three thousandyears of educational writing, see Ulich (1999).


What is learning?

Once you start to enquire into the nature of learning, it is amazing how hardit is to pin down what you mean. In every generation, authors have triedbecause in order to develop effective teaching, it is necessary to be able toidentify and verify learning. The extracts in the rest of this chapter begin witheducation in general, and then move on to focus on aspects with morespecific relevance to mathematics.

Education in general

Education: Alfred North Whitehead

Whitehead was concerned not only with learning, but also with teaching.Here he coins the expression inert knowledge (see also p. 288):

I appeal to you, as practical teachers. With good discipline, it isalways possible to pump into the mind of a class a certain quantity ofinert knowledge. You take a text-book and make them learn it. So far,so good. … But what is the point of teaching a child to solve aquadratic equation? There is a traditional answer to this question. Itruns thus: The mind is an instrument, you first sharpen it, and thenuse it; the acquisition of the power of solving a quadratic equation ispart of the process of sharpening the mind. Now there is just enoughtruth in this answer to have made it live through the ages. But for allits half-truth, it embodies a radical error which bids fair to stifle thegenius of the modern world. … The mind is never passive; it is aperpetual activity, delicate, receptive, responsive to stimulus. Youcannot postpone its life until you have sharpened it. Whateverinterest attaches to your subject-matter must be evoked here andnow; whatever powers you are strengthening in the pupil, must beexercised here and now; whatever possibilities of mental life your


Distinguishing cognition (awareness), affect (emotion) and enaction (behav-iour) is supposed to make it easier to make sure that all three are involved, inorder to make learning effective.

Reflection is best stimulated by awakening an internal monitor whichwatches and questions what we are doing while we are doing it.

Every generation, every educator tries to articulate their sense of whatlearning is, and how it can best be supported and made efficient and effective.There are deep roots for even the most modern and radical of proposals.

There are fundamental obstacles to learning; learners need help in over-coming these through psychologising of the subject matter.

teaching should impart, must be exhibited here and now. That is thegolden rule of education, and a very difficult rule to follow.

The difficulty is just this: the apprehension of general ideas, intellec-tual habits of mind, and pleasurable interest in mental achievement canbe evoked by no form of words, however accurately adjusted. All prac-tical teachers know that education is the patient process of the masteryof details, minute by minute, hour by hour, day by day. There is noroyal road to learning through an airy path of brilliant generalisations.There is a proverb about the difficulty of seeing the wood because ofthe trees. That difficulty is exactly the point which I am enforcing. Theproblem of education is to make the pupil see the wood by means ofthe trees.

(Whitehead, 1932, pp. 8–10)

Note the resonances between attributes you wish to evoke in the minds ofchildren and Dewey’s notion of psychologising the subject matter (see p. 45and p. 203), Freudenthal’s didactical phenomenology (see p. 202), andGattegno’s educating awareness (see p. 61 and 204) among others. White-head here also seems to be referring to a form of the transpositiondidactique (see p. 83).

Learning in general

What does it mean to know?: Gianbattista Vico

Gianbattista Vico (b. Naples, 1668–1744) was a philosopher scientist whoprobed what it means ‘to know’. His ideas have only relatively recently beentaken up and developed. He concluded that people can only know forcertain what they have made: other kinds of knowing are not certain in thesame way. When people make things with their hands they have objects thatthey can show to others to say ‘This is what I know.’ However, when peoplemake mental objects it is not so easy to offer them to others to check out or touse. Vico’s refrain was Verum ipsum factum: ‘the truth is the same as themade’; this can be expanded to ‘ … human truth is what man comes to knowas he builds it, shaping it by his actions’ (Vico, quoted in von Glasersfeld,1984, p. 27).

The aim of learning is to construct meaning for ourselves, not to attainexternal, pre-existent meanings. At the same time, we need to conform toagreed social practices. In education, this belief was most notably taken upby Dewey (see p. 45) who said that education should involve the use anddevelopment of what learners bring with them to their learning, thusdrawing a picture of active learners taking some kind of personal roles intheir construction of meaning.

Vico’s ideas were brought to the attention of the mathematics educationcommunity largely through the writing and scholarship of Ernst von


Glasersfeld who was seeking the roots of Piaget’s notion of the learner asconstructer of knowledge (genetic epistemology, p. 148).

Constructivism: Ernst von Glasersfeld

Ernst von Glasersfeld (b. Munich, 1917–) was at various times a student ofmathematics, journalist, and farmer before emigrating to the United Stateswhere he established a reputation amongst educationalists for his chal-lenging position as a radical constructivist (see p. 93), and as a philosopherand cybernetician. Building on both Piaget and Vico, von Glasersfeldextended and sharpened Piaget’s own form of constructivism, genetic episte-mology (see p. 148). His fluency in German, Italian and French led him tocriticise most translations of Piaget for their inaccuracy and misleadingnature. Here he summarises key aspects of his radical constructivism in rela-tion to Piaget and Vico.

From an explorer who is condemned to seek ‘structural properties’ ofan inaccessible reality, the experiencing organism now turns into abuilder of cognitive structures intended to solve such problems as theorganism perceives or conceives. … Piaget (1937) characterized thisscenario as neatly as one could wish: ‘intelligence organizes the worldby organizing itself’. What determines the value of the conceptualstructures is their experiential adequacy, their goodness of fit withexperience, their viability as means for solving of problems, amongwhich is, of course, the never-ending problem of consistent organiza-tion that we call understanding.

The world we live in, from the vantage point of this new perspective,is always and necessarily the world as we conceptualize it. ‘Facts’, asVico saw long ago, are made by us and our way of experiencing, ratherthan given by an independently existing objective world.

(von Glasersfeld, 1983, pp. 50–1)

There are similarities with Bartlett, Hanson and Goodman (see p. 31). WhileDewey and many others had stressed the need for learners to be active,Piaget was claiming that learners are necessarily active, even if they are notactively making the sense that the teacher expects or intends them to make.Von Glasersfeld pushed that further to suggest that there is no way in whichthe existence of an objective external reality can be affirmed. Rather, allobservers have to go on is their sense of ‘fit’ between what they conjectureand what they construe from their sense impressions. For example, once alesson is finished, a teacher cannot say that there was a single lesson, butrather the lesson-event consists of the stories told by each participant inthinking back over it.


Learning as making distinctions

We turn now to more specific and mathematically pertinent views oflearning. The root awareness, which all babies have even before they areborn, is the awareness of distinction making, for example, distinguising thesound of mother’s voice from other noises. Distinction making is perhapsthe most fundamental power possessed by all sensate beings (including allanimals and plants). Differences between organisms arise in the difference(there it is again: making distinctions) between the range and refinement ofdistinctions made. Distinction-making often creates tension, which is aform of disturbance, prompting further exploration of relationships, prop-erties and so on, all in an attempt to re-integrate the distinguished andappreciate the whole.

Discernment: Ference Marton

Ference Marton (b. Hungary, 1939–) moved eventually to Sweden wherehe has led a long programme of research at all levels and in all subjects. Heand his colleagues also developed the approach to research known asphenomenography (Marton, 1981), which charts (-graphy) the variations inlived experience (phenomena) of learners as opposed to phenomenology,which is a well-established philosophical movement.

… learning takes place, knowledge is born, by a change in something inthe world as experienced by a person. The new way of experiencingsomething is constituted in the person–world relationships and involvesboth. … Person and world, the inner and outer are not separated. We donot have to account for how knowledge travels from one to the other.Instead of trying to account for how the person–world relationship isestablished, we posit this relationship and study how it changes as timepasses.

(Marton and Booth, 1997, p. 139)

… teachers mold experiences for their students with the aim of bringingabout learning, and the essential feature is that the teacher takes the partof the learner, sees the experience through the learner’s eyes, becomesaware of the experience throughout the learner’s awareness. If we …consider the learner to be internally related to the object of learning, andif we consider the teacher to be internally related to the same object oflearning, we can see the two, learner and teacher, meet through a sharedobject of learning. In addition to this, the teacher makes the learner’sexperience of the object of learning into an object of her own focalawareness: the teacher focuses on the learner’s experience of the objectof learning.

(ibid., p. 179)


In the next extracts, Marton and Booth focus on the fundamental act ofdiscerning foreground from background, and emphasise that making distinc-tions is the basis of learning. What you cannot discern, you cannot see.

In order to experience something as something we must be able todiscern it from and relate it to a context, and be able to discern its partsand relate them to each other and to the whole. But we discern wholes,parts, and relationships in terms of aspects that define the wholes, theparts, and the relationships. To discern the spatial arrangement of thelandscape we have to experience the spatial arrangement as a spatialarrangement (among other conceivable spatial arrangements).

(ibid., p. 108)

… the act of gaining knowledge about the world involves qualitativedifferences … in the way things are experienced (understood, conceptu-alized, apprehended, etc.) – as phenomena, situations, or learning itself.The question that now becomes interesting is that of what it means andwhat it takes to experience something in a particular way …

(ibid., p. 86)

The aspects of the phenomenon and the relationships between themthat are discerned and simultaneously present in the individual’s focalawareness define the individual’s way of experiencing the phenom-enon. Being focally aware of the weight of the body immersed in somefluid as compared to its weight when not immersed, of the fact that acertain volume of the fluid is displaced by the act of immersion, of theweight of the fluid displaced – all at the same time – amounts to what ittakes to discover, or to understand, Archimedes’ principle. The keyfeature of the structural aspect of a way of experiencing something(and thereby also of the referential aspect with which the structuralaspect is intertwined) is the set of different aspects of the phenomenaas experienced that are simultaneously present in focal awareness.

(ibid., p. 101)

For example, discerning the vertices and edges of a figure instead of just theoverall shape, and discerning the positions of digits in a numeral as repre-senting powers of ten and not just a string of numerals, alter how you see,what you can do, and the choices available to you.

Dimensions-of-variation: Ference Marton and Shirley Booth

Marton and Booth considered what learners are learning to discern in atypical lesson. They were thinking in terms of concepts, and the use andmeaning of technical terms.


To experience a particular situation in terms of the general aspects, wehave to experience the general aspects. These aspects correspond todimensions of variation. That which we observe in a specific situationwe tacitly experience as values in those dimensions. A certain way ofexperiencing something can at best be understood in terms of thedimensions of variation that are discerned and are simultaneously focalin awareness, and in terms of the relationships between the differentdimensions of variation.


If we notice that something is the case (e.g., there are 7 marbles hiddenin one box), the variation is implicit; there is an implication that therecould have been 6 or 8 or some other number of marbles, but we do notfocus on it. The dimensions are discerned in relation to the thematic fieldagainst the background of which the phenomenon, and the situation inwhich it is embedded, is seen.

(ibid., p. 101)

Learning is seen as learning to discern, which requires simultaneity–discern-ment–variation. See also Marion Walter’s work on ‘what-if’ questions (Brownand Walter, 1993).

Dimensions-of-possible-variation: Anne Watson and John Mason

Anne Watson (1948–) has taught in secondary schools in England, beenhead of department, and been a teacher educator at both primary andsecondary level. Her focus is always on mathematics, and she works topromote inclusive mathematics learning and teaching. With FerenceMarton, together with his student Ulla Runesson and John Mason, Anne hasextended the notion of dimension-of-variation to recognise that within anyone dimension-of-possible-variation different people might be aware ofdifferent permissible choices. For example, in an identity such as (x – 1)(x +1), some people, aware that x can take ‘any’ value, might be thinking onlyin terms of x as an integer, or as a rational number, or as a real number, oras a complex number, or as a square matrix, or as a differential operator …. Another dimension-of-possible-variation is the exponent 2, leading tofactoring of x

n– 1, and again people may be aware of different ranges of

permissible change for n (see also Stein, p. 177). A third dimension-of-possible-variation is the constant term 1 with a range-of-permissible-change being the nth power of some term. Thus tasks that reveal thedimensions-of-possible-variation, and the corresponding range-of-permis-sible-change in each dimension of which learners are aware, are pedagogi-cally useful. They indicate where further work may be required. SeeWatson and Mason (in press).


A single exercise or problem is an example of a task. Asking yourself whatcould be changed, while using the same approach or technique, opens updimensions-of-possible-variation; asking what sorts of change can be made inany one such dimension opens up range-of-permissible-change, leading tothe notion of variable and parameter. Together these two ideas transform a setof exercises from a sequence of tasks for learners to work through piecemeal,into a domain of generality – a technique in the full sense of the word.

Whenever learners see through a particular mathematical object to a moregeneral class of objects, or through a particular or mathematical exercise to aclass of problems, they are demonstrating awareness of a dimension-of-possible-variation. There are similarities with Whitehead on generalising(see p. 132) and also with the van Hiele phases (see p. 59) and the structureof attention (see p. 60).

Learning as structuring attention

The terms attention and awareness are used differently by different authors,often even interchangeably. We start with Marton and Booth who useawareness in much the way that others use attention.

Structuring awareness: Ference Marton and Shirley Booth

In Marton and Booth’s work on the anatomy of awareness (Marton andBooth, 1997), it is posited that a person’s way of experiencing somethingis related to how their awareness is structured. There is both a ‘what’aspect, which corresponds to the object of the experience, and a ‘how’aspect, which relates to the act of experiencing. The way of experiencingcan be couched in terms of a dynamic relationship between the twoaspects of the phenomenon, the structural aspect and the referential (ormeaning) aspect.

If we consider an individual at any instant, he or she is aware of certainthings or certain aspects of reality focally while other things havereceded to the background. But we are aware of everything all the time,even if not in the same way all the time. The structure of an individual’sawareness keeps changing all the time, and that totality of experience iswhat we call the individual’s awareness. An experience is an internalrelationship between the person experiencing and the phenomenonexperienced: it reflects the latter as much as the former. If awareness isthe totality of all experience, then awareness is as descriptive of theworld as it is of the person. A person’s awareness is the world as experi-enced by the person.


There are similarities with stressing and ignoring (p. 127).


Structuring attention: Van Hiele and Van Hiele

Dina and Pierre van Hiele (b. The Netherlands) studied the development ofgeometrical thinking, particularly in school. Dina was an experiencedteacher of mathematics who was notable (see p. 163) for studying her ownteaching, but died tragically young; Pierre is the theorist who then extendedtheir work beyond geometry.

The clearest exposition of the different types of thinking that the vanHieles discerned can be found in later authors (for example, Burger andShaughnessy, 1986) who tried to develop tests to distinguish the different‘levels’ of thinking. However, the van Hiele ‘levels’ were intended todescribe thinking in the moment, not to categorise learners. At any givenmoment a person can be looking at one aspect holistically but withoutdiscerning features, while being aware of other properties and reasoningquite formally with still others. The following descriptions of the van Hielelevels have been modified by replacing ‘a student at this level’ (whichimplies that people can be classified as at certain levels) by ‘functioning atthis level’ (which implies that at different times people may functiondifferently).

• Level 1 (Visualisation) Functioning at this level involves reasoningabout basic geometric concepts, such as simple shapes, primarily bymeans of visual considerations of the concept as a whole withoutexplicit regard to the properties of its components. One ‘sees’, forexample, that two triangles are similar, without recourse to reasons oraspects (see also van Hiele, 1986, p. 83).

• Level 2 (Analysis, description using language developed at previouslevel) Functioning at this level involves reasoning about geometricconcepts by means of an informal analysis of component parts andattributes. Necessary parts of the concept are established. For instance,similar triangles can be recognised and justified in terms of possession ofproperties (see also van Hiele, 1986, pp. 83–4).

• Level 3 (Abstraction, using language for distinctions developed atprevious level) Functioning at this level involves thinking of, say, a rect-angle as a collection of properties that it must have (necessary condi-tions). When asked why a figure is a rectangle, the response would bea litany of properties: ‘Opposite sides are parallel, opposite sides arecongruent, opposite angles are equal, you have four right angles … ’.Likewise, similar triangles can be located and justified by recourse tonecessary conditions, using an argument involving statementsfollowing from or depending on each other in sequence (see also vanHiele, 1986, p. 84).

• Level 4 (Informal deduction) Functioning at this level means selectingsufficient conditions from the ‘litany’ described in Level 3 to determine arectangle. Properties are ordered logically and the role of general


definitions is appreciated. Simple inferences can be made, and classinclusions are recognised (for example, squares are rectangles).

• Level 5 (Rigour, Formal deduction) Functioning at this level meanscomparing systems based on different axioms, and studying variousgeometries in the absence of concrete models.

(adapted from Burger and Shaughnessy (1986))

In van Hiele terms, Level 3 awareness of a geometric figure means awarenessof a variety of properties, each of which constitutes a relationship that holdsbetween perceived details of the figure. However, connections betweenthese properties are at best dimly sensed, and the properties are seen aspertaining to the particular geometrical object rather than independent of it.Thus, angles, edge-lengths, triangles and circles may be discerned, but notthe necessary links between them. The notion that some properties are suffi-cient to force a figure to have a certain quality such as being a square or aparallelogram is part of the fourth level awareness. Recognising equality ofangles as implying similarity and so being led to look at ratios of corre-sponding lengths shows logical organisation of properties, hence level fiveawareness.

The SOLO taxonomy (see p. 309) has some similarities with van Hielephases, as does Alan Bell’s principles of task design, and also the onionmodel of understanding (see p. 298).

There is a slightly different way of thinking about phases, which corre-sponds closely to that of van Hiele but which was derived quite independentlyfrom ancient sources (Bennett, 1956–1966; 1993). It focuses on what learnersare attending to at any moment, and describes this in terms of the structure ofattention:

• attention on the whole, the global;• attention on distinctions, distinguishing and discerning aspects, detailed

features and attributes;• attention on relationships between parts or between part and whole,

among aspects, features and attributes discerned;• attention on relationships as properties that objects like the one being

considered can have, leading to generalisation;• attention on properties as abstracted from, formalised and stated independ-

ently of any particular objects, forming axioms from which deductions canbe made.

There are also important shifts from seeing something in its particularity toseeing it as representative of a general class.

For distinction-making as a natural human power, see p. 124; for its biologicalbasis see later in this section (p. 70).


Experiencing the world: Ingrid Pramling

Ingrid Pramling (b. Sweden, 1946–) was a research student of FerenceMarton. Here she describes in the abstract of her thesis the approach that shetakes to working with young learners:

How children conceptualise, experience, discern, see, understand theworld around them is the ground from which their skills and knowledgeoriginate. This implies that early childhood education should primarilyfocus on the different ways children are capable of being aware of thevarious phenomena in the world around them. Above all, we shouldfind out the critical, and usually taken for granted, aspects of our ways ofexperiencing the world around us on which our capabilities for dealingwith the world rest. Learning from this perspective is learning to experi-ence the world in particular ways.

(Pramling, 1994)

Pramling has developed this view into the importance of reflection (see p. 286).

Learning as educating awareness

Educating awareness: Caleb Gattegno

Caleb Gattegno (1911–1988) grew up in Alexandria. His lifelong concern was todevelop a systematic enquiry into human learning. He moved to England wherehe was involved in teacher education, founding what later became the Associa-tion of Teachers of Mathematics and co-founding the International Commissionfor the Study and Improvement of Mathematics Teaching (CIEAEM), an interna-tional body, which has an annual conference in Europe. He discovered Cuise-naire’s ‘rods’, which he developed and promulgated around the world, makingfilms of demonstration lessons and publishing books about their use. He thenmoved into teaching foreign languages, using what he called The Silent Waybased on the use of colours to represent sounds. Gattegno went on to articulatea science of education based on close analysis and study of the universe ofbabies and how learning takes place in the first few years.

Observation of very young babies yields evidence that in crying, forinstance, there is the presence of consciousness and that crying is notonly the use of reflex mechanisms but also reveals the presence of thecrier in the cries. Only the presence of consciousness explains thereason these cries can be made louder, more prolonged, and can bestructured at will to express small things the baby knows intimately frominside. At four weeks, for example, a baby may find in his or her crysounds which have attributes that can be extracted from the whole,singled out for special attention and examined per se before the baby


returns to crying and the purpose for which the crying was started and isbeing continued.

Perhaps one must have encountered a phenomenon like this to allowoneself to entertain as an instrument of study the multiple dialogues theself has with itself, even from conception.

(Gattegno, 1987, p. 7)

When someone becomes aware that they can establish a one-to-one corre-spondence between marks on wood and sheep, say, then the act of countingbecomes available for study, whereas previously it had only been an actionto undertake. This for Gattegno is the source of all disciplines: peoplebecoming aware of the actions they perform and turning these into an objectof study: aware of their awareness.

Further levels are generated as people move into teaching, and then intoteacher education (see Mason, 1998). There are similarities with theorems-in-action (see p. 63), and with enactivism (see p. 70).

Gattengo’s notion of teaching as arranging for learners to educate theirawareness is extended to include training in behaviour through harnessingof emotions (see p. 204).

Awareness-in-action

As has already been pointed out, there are things that people are not awareof but which are vital to how they function, and things that they have beenaware of but no longer need to be. The latter can be referred to as aware-ness-in-action. For example, people can count without being aware of one-to-one correspondence, and people can add, subtract, multiply, and divide,without being explicitly aware of their awarenesses-in-action of numerals,place-value, routines, the role of order, and so on, which makes that arith-metic possible. People can form and detect patterns and locate formulae thatgeneralise specific cases without being explicitly aware of their awarenesses-in-action of same and different, relatedness, induction, stressing andignoring. People can combine fractions according to rules, without beingaware of how fractions relate to decimals and to integers, and how theygeneralise number, without being aware of the slide between operator andobject and without relating them to a number-line.

Awareness therefore has different forms, here exemplified by the notion ofangle:

• awarenesses-in-action: things you can do with an angle, but cannotarticulate or even be consciously aware of (but which are consonantwith what later become theorems);

• awarenesses of sense: things which are pre-articulate, yet somehowpresent in consciousness and informing action: a sense of big angles andof rotation;


• awarenesses in articulation: things you can say about angles, not just dowith them, and even things you know but find hard to do physically.

Vergnaud (1981) referred to awarenesses-in-action as theorems-in-action,while awarenesses-in-articulation are conscious theorems.

Awareness is not everything though. Awareness is based on actions, andleads to further actions. Behaviour, whether physical, mental, symbolic orvirtual develops as a result of becoming aware of existing awarenesses.Awareness alone is like knowledge without skill: nothing actually happens.Learning involves both educating awareness and using that awareness todirect behaviour, which can be trained by developing habits.

Awareness and behaviour together are not sufficient to characteriselearning, because something is needed to produce action – to make thingshappen. The source of energy lies in the emotions. Emotions are what haveto be harnessed in order to engage in activities through which learning cantake place. This view is captured in the image of the human psyche as achariot or carriage (see p. 33). The notions of discipline, of being systematic,of concentrating and persisting, all depend on emotional energy. If you arefeeling unconfident or if you feel unvalued, then attention and desire can begreatly attenuated. Emotions come into play when people become aware ofa disturbance (see p. 55 and p. 101).

Learning as developing intuition

Awareness and intuition are closely related though not identical.

Intuition: Efraim Fischbein

Efraim Fischbein (b. Romania, 1920–1998) was an Israeli mathematicseducator and researcher. He was the founding president of the internationalgroup Psychology of Mathematics Education (PME), which runs an annualinternational conference. As a researcher he was particularly interested in therole that intuition plays, both in problem solving and in learning newconcepts, and he is best known for his seminal work on intuition in scienceand mathematics. More recently he studied the role of diagrams in teaching.

Intuition needs to be set against formal deduction, which so many peoplebelieve characterises mathematics, even though mathematicians themselvessee deductive proof as merely the final step, the wrapping up in order toconvince others. The creative aspects of mathematics lie in bringing toexpression nascent and developing intuitions.

There is today much evidence – both experimental and descriptive – thatno productive mathematical reasoning is possible by resorting only toformal means. One may possess all the formal knowledge relevant to amathematical topic (definitions, axioms, theorems, proofs, etc.) and yet


the system does not work by itself in a productive manner (for solvingproblems, producing theorems and proofs, etc.).

(Fischbein, 1987, p. 16)

Our theory is that mental behavior (reasoning, solving, understanding,predicting, interpreting) including mathematical activity, is subjected tothe same fundamental constraints. The mental ‘objects’ (concepts, opera-tions, statements) must get a kind of intrinsic consistency and directevidence similar to those of real, external, material objects and events, ifthe reasoning process is to be a genuinely productive activity.

(ibid., p. 20)

Here Fischbein considers how mathematics relates to the material world andthe world of symbols. His views are challenged by those who see mathe-matics as necessarily fallible and social.

This world of constructs – the world of mathematics – seems to mirror allthe features which enable the known real world to function. To be sure,it mirrors them on its own terms, but all the ingredients seem to exist forconferring credibility, consistency, coherence, on this world of mentallyproduced abstractions.

In other words, the human mind seems to have learned from the basicgeneral properties of empirical reality how to build an imaginary, struc-tured world, similarly governed by rules and similarly capable of consis-tency and credibility. The fundamental difference is that in the empiricalworld the constraints (invariant properties and relationships) are implic-itly given, while in the formal world every property and every relation-ship is stated and justified explicitly. The history of mathematics is thehistory of the human endeavor for shaping a new type of certitudedealing with explicitly postulated entities governed by explicitly,formally-stated rules.

… the ideal aim of this endeavor has been, and still is, the creation of aworld of concepts which may function coherently in an absolutelyautarchic way. This was the dream of the modern, axiomatic approach.

(ibid., p. 16)

Mathematics is perhaps unique in formalising concepts in succinctly manipu-lable symbols that stand for mathematical objects (see reification, p. 167). Thisleads to problems of how to explain the effective interaction between thesethree worlds – the worlds of concepts, symbols and objects. How can ideasinfluence the material world? How is it that the symbols and equations of mathe-matics so effectively enable the prediction of events in the material world whenmany of those ideas go well beyond anything that can be found in the materialworld (for example, infinite decimals, complex numbers, and so on). This‘unreasonable effectiveness’ was marvelled at by Eugene Wigner (see p. 194).


An intuition is, then, an idea which possesses the two fundamentalproperties of a concrete, objectively-given reality; immediacy – that is tosay intrinsic evidence – and certitude (not formal conventional certitude,but practically meaningful, immanent certitude).


We consider that the emergence of apparently self-evident, self-consis-tent cognitions – generally termed intuitions – is a fundamental condi-tion of a normal, fluent, productive reasoning activity. An intuition is acomplex cognitive structure the role of which is to organize the availableinformation (even incomplete) into apparently coherent, internallyconsistent, self-evident, practically meaningful representations.

(ibid., p. 211)

In the last extract, one might wish to replace ‘certitude’ with ‘strongconjecture’!

The notion of where and whether mathematical objects and intuitions‘exist’ has haunted mathematical philosophers ever since Plato decided thatthey belong to an independent world of forms, accessed through mentalimagery. Fischbein was not so concerned about where mental objects exist:

Mathematical entities such as numbers and geometrical figures do nothave an external, independent existence as the objects of the empiricalsciences do. In mathematics, we deal with entities whose properties arecompletely fixed by axioms and definitions. Dealing with such entitiesrequires a mental attitude which is fundamentally different from thatrequired by empirical, materially existing realities.

(ibid., p. 206)

In terms of van Hiele levels (see p. 59), Fischbein is looking at the need tobecome aware of properties independent of particular objects, and howdeductions can be made from those properties alone on the basis of conjec-tures arising from familiarity with one or more examples. This leads todidactical challenges:

An essential recommendation is to create didactical situations which canhelp the student to become aware of … conflicts. However, renderingmanifest the latent conflicts does not solve the problem by itself. Thisprocedure has to be associated with the already mentioned activity ofanalyzing explicitly the properties – as stated by definitions – of themathematical entities considered (in contrast with the intuitiveinterpretations).

[ … ]… initial intuitive interpretations become very strongly attached to the

respective concepts and, consequently, it becomes very difficult to


escape their impact. Yet it is impossible to avoid using intuitive meansinitially when introducing new mathematical concepts. While there is nogeneral recipe for solving the dilemma, this is what we recommend. Onehas to start, as early as possible, preparing the child for understandingthe formal meaning and the formal content of the concepts taught. Thismay be done, first of all, by revealing the relationships betweenconcepts and operations and by rendering explicit the underlyingcommon structures of different concepts and operations. Multiplicationand division, for instance, are inverse aspects of multiplicative structureswhich, in turn, are related to proportional reasoning.

[ … ]… it is important to develop in students the conviction that: (a) one

possesses also correct, useful intuitions and (b) that we may becomeable to control our intuitions by assimilating adequate formal structures.

[ … ]When referring to the development of intuitions one has to consider

also anticipatory intuitions. Though mathematics is a deductive systemof knowledge, the creative activity in mathematics is a constructiveprocess in which inductive procedures, analogies and plausible guessesplay a fundamental role. The effect is very often crystallized in anticipa-tory intuitions. Much more attention should be given, in our opinion, toeducating students’ sensibility for similarities, the ability to identify …and describe, structures.

(ibid., pp. 208–10)

Compare the first paragraph in the extract above with the role of distur-bance, surprise and Festinger’s cognitive dissonance (see p. 69). There aresimilarities with didactic phenomenology (p. 202) and awareness of aware-ness (p. 186 and p. 189).

The issue of intuition, as opposed to deductive rigour, and of objectiveexistence of mathematical objects in some ‘world’, leads to the need todistinguish between what is arbitrary (decided by convention) and what isnecessary (see Hewitt, p. 159).

Beyond practically defined circumstances [the concept of a straight line]is a convention defined in the frame of a certain group of axioms, whichmay be changed. But via extrapolation (from a behavioral meaningcontextually dependent on an absolute universal concept), one tends tobelieve in the absoluteness of the concept, one tends to confer on thisnotion, based on conventions, the absoluteness of a given, objectivelyexisting fact. In our terminology, this means conferring intuitivity on theconcept, or, in other words, the concept of a straight line gets an intu-itive meaning for the individual.

(ibid., p. 21)


Intuitivity is closely related to reification (see p. 167). Fischbein concludeswith:

It is important to emphasize that new, correct intuitions do not simplyreplace primitive, incorrect ones. Primary intuitions are usually so resis-tant that they may coexist with new, superior, scientifically acceptableones. That situation very often generates inconsistencies in the student’sreactions depending on the nature of the problem.

[ … ]In order to cope successfully with the instructional problems, one has

first to have a good, serious understanding of the psychological aspectsof the concepts involved. Much more research is needed in this respect.We have to know what are the tacit interpretations the student attachesto these concepts, what are the intuitive reactions, the intuitive modelshe produces, the impact they may have on the acquisition of the newconcepts. On the other hand, one has to evaluate the effect of thevarious didactical means on the complex and labile relationshipsbetween the intuitive loading of the concepts taught and their formalstructure. An inadequate strategy may destroy the productive interactionof these two components.

(ibid., pp. 213–14)

Fischbein taught that incomplete or inappropriate intuitions are alwayspresent, but hopefully overlaid with educated intuitions. Also that throughan appropriate classroom atmosphere and ethos, learners can develop intu-itions concerning gaps in arguments, dissonances between different ways ofthinking about or describing situations, and so on.

Note the similarities between Fischbein’s reflections and Tall and Vinner’snotion of concept-image (see p. 200).

Learning as response to disturbance

Virtually every educational theory acknowledges that organisms (incudinghumans) respond to disturbances from their environment. Some theoriesmake more of this than do others.

Discrepancies lead to discovery: Theodoret Cook

This extract is taken from the preface of Cook’s book The Curves of Life(Cook, 1979) in response to heavy criticism of his conjectural notebooks inwhich he had used mathematical ideas to model pure form from whichmanifestations in nature seemed to deviate.

It is largely by noting approximations … and then investigating the devi-ations that ‘knowledge grows from more to more’. Discrepancies lead to


Discovery. … deviations are one cause of beauty and one manifestationof life; and this is why the study of exceptions is the road to progress.

(Cook, 1979, preface)

Attending to what learners are struggling to express, and interacting withthem on that basis, can lead to more open and less limiting forms of ques-tioning. However, attending to deviation from an ideal is at the heart of thelanguage of learners’ errors and misconceptions (see p. 212), and leadsnaturally to forms of questioning that are limiting, and even to funnelling(see p. 274).

Disturbance: Jerome Bruner

Bruner proposed that human beings are essentially narrative animals, tellingstories to themselves and others as a way of making sense of the world. Herehe is analysing the components of a narrative, and discovers that at the heartof a good narrative is some sort of disturbance.

At a minimum, a ‘story’ (fictional or actual) involves an Agent who Actsto achieve a Goal in a recognizable Setting by the use of certain Means.What drives the story, what makes it worth telling, is Trouble: somemisfit between Agents, Acts, Goals, Settings, and Means. Why is Troublethe license for telling a story? Narrative begins with an explicit or implicitprologue establishing the ordinariness or legitimacy of its initial circum-stances – “I was walking down the street minding my own businesswhen … ”. The action then unfolds leading to a breach, a violation oflegitimate expectancy. What follows is either a restitution of initial legiti-macy or a revolutionary change of affairs with a new order of legitimacy.Narratives (truth or fiction) end with a coda, restoring teller and listenerto the here and now, usually with a hint of evaluation of what hastranspired.

(Bruner, 1996, pp. 94–5)

Note the parallels with assimilation and accommodation (see p. 149), recasthere as the basis for the construction of narrative. People ‘wake up’ whenthere is some sort of disturbance from the expected flow; the same applies tolearners’ experiences in lessons. There is more likely to be some meaning-making through the construction of personal narratives based on sharedclassroom and peer group narratives if there is something that surprises ordisturbs, something that catches attention.

Learner narratives include purposes for specific activities and for schoolitself, as well as for how a technique is used, and why.

… what is [important] is the procedure of the inquiry, of mind using,which is central to the maintenance of an interpretive community and a


democratic culture. One step is to choose the crucial problems, particu-larly the problems that are prompting change within our culture. Let thoseproblems and our procedures for thinking about them be part of whatschool and classwork are about. This does not mean that school becomesa rallying place for discussion of the culture’s failures. … Trouble is theengine of narrative and the justification for going public with the story. Itis the whiff of trouble that leads us to search out the relevant or respon-sible constituents in the narrative, in order to convert the raw Trouble intoa manageable Problem that can be handled with procedural muscle.

(ibid., pp. 98–9)

Choosing fundamental problems for learners to work on over a period oftime is one common approach to curriculum development (see also investi-gative teaching, p. 225).

Cognitive dissonance: Leon Festinger

Leon Festinger (b. New York, 1919–1989) was a social psychologist inter-ested in human learning. He is best known for his theory of cognitive disso-nance, which can be used to account for certain observations that mostteachers will have experienced, if not marked and remarked upon for them-selves: it requires some sort of a disturbance, some sort of surprise or brokenexpectation to capture attention and initiate enquiry.

1 The existence of dissonance, being psychologically uncomfortable,will motivate the person to try to reduce the dissonance and achieveconsonance.

2 When dissonance is present, in addition to trying to reduce it, theperson will actively avoid situations and information which wouldlikely increase the dissonance.

[ … ]… dissonance, that is, the existence of nonfitting relations among

cognitions, is a motivating factor in its own right. … Cognitive disso-nance can be seen as an antecedent condition which leads to activityoriented toward dissonance reduction just as hunger leads to activityoriented toward hunger reduction.

(Festinger, 1957, p. 3)

Festinger suggested a number of different sources for dissonance, including:local and global inconsistencies; cultural mores; past experience; rewards orthreats for behaviour at variance with personal opinion; and, after a decision,when positive features of the rejected alternative, and negative features ofthe chosen one, become salient.

Festinger went on to make conjectures about the effects of differingmagnitudes of dissonance and consonance. Alan Bell’s diagnostic teaching


(p. 233) is particularly notable for making use of the role of cognitive disso-nance as disturbance in the design of teaching.

Disturbance is often experienced as surprise (see Moshovits-Hadar, p. 101),and this is a component that is central component to motivation and theharnessing of emotions (see p. 55 and p. 204). When an organism respondsto disturbance in its environment, the disturbance leads to action, and actionproduces learning.

Learning as action

Enactivism: Humberto Maturana and Francisco Varela

A rather different perspective on learning arises from the writings ofHumberto Maturana (1928–), who principally researches the biological basesof communication, extending his research into many different domains. Hetakes a view referred to as enactivism in which ‘action is knowledge andknowledge is action’.

There are similarities with Piaget’s remark ‘intelligence organises the worldby organising itself’ (see p. 94).

An observer is a human being, a person, a living system who can makedistinctions and specify that which he or she distinguishes as a unity, asan entity different from himself or herself that can be used for manipula-tions or descriptions in interactions with other observers. An observercan make distinctions in actions and thoughts, recursively, and is able tooperate as if he or she were external to (distinct from) the circumstancesin which the observer finds himself or herself. Everything said is said byan observer to another observer, who can be himself or herself.

(Maturana, 1978, p. 31)

The last statement may seem rather obvious at first, but it has profoundimplications, for it asserts that any form of speech is second order: it is notdirect experience but experience recast into language, as if from an observer.There are significant implications for this perspective. For example:

An explanation is always a proposition that reformulates or recreates theobservations of a phenomenon in a system of concepts acceptable to agroup of people who share a criterion of validation.

(Maturana and Varela, 1988, p. 28)

This fits well with conclusions drawn by Hanson and by Goodman amongmany others (see p. 31), underlying the discipline of noticing (Mason,2002c), about the intricate intertwining of observation and validity, and theview that there are no objective criteria for ‘truth’.


Maturana goes on to elaborate:

… we can distinguish four conditions essential to proposing a scientificexplanation. They do not necessarily fall in sequential order but dooverlap in some way. They are:

a describing the phenomenon (or phenomena) to be explained in away acceptable to a body of observers;

b proposing a conceptual system capable of generating the phenom-enon to be explained in a way acceptable to a body of observers(explanatory hypothesis);

c obtaining from (b) other phenomena not explicitly considered in thatproposition, as also describing its conditions for observation by abody of observers;

d observing these other phenomena obtained from (b).(Maturana and Varela, 1988, p. 28)

Maturana’s writing slides quickly from the pragmatic to the ethereal. Here heand his colleague Francisco Varela (b. Chile, 1946–2001) reject some of thedistinctions others make, say, between different worlds (see p. 73), and thenaddress the issue of ‘observation’ from a researcher’s or a teacher’s point ofview: what can we see when we observe?

… our experience is moored to our structure in a binding way. We donot see the ‘space’ of the world; we live in our field of vision. We do notsee the ‘colors’ of the world; we live our chromatic space. Doubtless …we are experiencing a world. But when we examine more closely howwe get to know this world, we invariably find that we cannot separateour history of actions – biological and social – from how this worldappears to us. It is so obvious and close that it is hard to see.

(ibid., p. 23)

This is entirely compatible with von Glasersfeld’s radical constructivism (seep. 93), and with Goodman’s and Hanson’s views (see p. 31).

… underlying everything we say is this constant awareness that thephenomenon of knowing cannot be taken as though there were ‘facts’or objects out there that we grasp and store in our head. The experienceof anything out there is validated in a special way by the human struc-ture, which makes possible ‘the thing’ that arises in the description.

This circularity, this connection between action and experience, thisinseparability between a particular way of being and how the worldappears to us, tells us that every act of knowing brings forth a world. …All doing is knowing, and all knowing is doing.

(ibid., p. 25–6)


The notion of ‘bringing forth a world’ aptly describes experience as someonelaunches into speech and others try to make sense of what is said. Thisapplies specifically to mathematics, where a teacher is comfortably navi-gating a world brought forth by some stimulus, but learners bring forth onlythe worlds available to them.

Several authors have taken up the enactivist view. For example, there is aclose agreement with Greeno’s environment view of mathematical learning,and with Cobb, Yackel and Wood’s enriched constructive view of classroomlife that may help situate them:

In the environmental view, knowing a set of concepts is not equivalentto having representations of the concepts but rather involves abilities tofind and use the concepts in constructive processes of reasoning. … Theperson’s knowledge … is in his or her ability to find and use theresources, not in having mental versions of maps and instructions as thebasis for all reasoning and action.

(Greeno, 1991, p. 175)

We emphasize that mathematics is both a collective human activity …and an individual constructive activity.

(Cobb et al., 1992, p. 17)

Enactivism: Sen Campbell and Sandy Dawson

Sandy Dawson (b. Alberta, 1940–) is a teacher and teacher educator inspiredby Caleb Gattegno and by Humberto Maturana. Sen Campbell (b. Alberta,1952–) is also a strong advocate of enactivism. Both Dawson and Campbellsee the classroom in terms of stressing and ignoring, whether by the learneror by the environment:

… the learner chooses to stress certain aspects of the activities and toignore others. It is not just a one-way street, however, with the learnerbeing the predominant force. The environment exerts itself by puttinglimits on what pathways the learner is able to pursue. Learning occurs atthe interstices where the learner meets the environment, stresses particu-larities within that environment, and generates a response whoseviability in the environment is then determined. However, the realm ofthe possible must intersect with the predilection of the learner to takenotice of it. The particular pathway mutually determined by the learnerand the environment is rarely unique. Other pathways are possible. Theone selected is but one of many ways of satisfying the demands of theinteraction as seen by the learner and permitted by the environment.

(Campbell and Dawson, 1995, p. 244)

There is similarity with affordances (p. 246).


… the learning which teachers hope will occur does not happen simplyas the result of engaging learners in particular activities, whether thoseactivities are, for example, a teacher talking to/with learners, learnerswatching a film, or learners using a computer … . however, if learnersdon’t get it, (whatever it might be), it doesn’t follow that there is some-thing wrong with the learners, or that the learners weren’t listening, orweren’t paying attention. It is just that the learners were at that point intime stressing things other than those which the teacher might haveanticipated, or that environment and those learners – given their partic-ular history of structural coupling – did not opt for the pathway expectedby the teacher. It is evident, then, that teaching is not telling.

(ibid., p. 245)

See also epistemological obstacles (p. 303).

Radical enactivism: Brent Davis, Dennis Summara and Tom Kieren

A more radical perspective on enactivism sees learning and action as oneand the same (see also Maturana, p. 70):

Learning should not be understood in terms of a sequence of actions,but in terms of an ongoing structural dance – a complex choreography– of events which, even in retrospect, cannot be fully disentangled andunderstood, let alone reproduced.

(Davis, Summara and Kieren, 1996, p. 153)

Three worlds (material, imagined, symbolic): Jerome Bruner

Where do the actions take place that are so strongly espoused as the essenceof learning? Immersed as we are in the material world, philosophers fromPlato and before were aware that human beings, at least, occupy themselveswith an inner world of mental images, ideas, and concepts, and a mediatingworld of symbols that afford access to images and that express images.

… a view of human beings who have developed three parallel systemsfor processing information and for representing it – one through manip-ulation and action, one through perceptual organization and imagery,and one through symbolic apparatus. It is not that these are ‘stages’ inany sense; they are rather emphases in development. You must get theperceptual field organized around your own person as center beforeyou can impose other, less egocentric axes upon it, for example. In theend, the mature organism seems to have gone through a process of elab-orating three systems of skills that correspond to the three major toolsystems to which he must link himself for full expression of his


capacities – tools for the hand, for the distance receptors, and for theprocess of reflection.

(Bruner, 1966, p. 28)

The fundamental triad: Anna Sierpinska

Anna Sierpinska (b. Poland, 1947–) moved to Canada where she is a leadingresearcher and mathematics educator.

[Hall (1981)] speaks of the ‘fundamental triad’: three levels of experi-encing the world by man, three ways of transmission of this experienceto children, three types of consciousness, three types of emotional rela-tions to things: the ‘formal’, the ‘informal’, the ‘technical’.

The ‘formal’ level is the level of traditions, conventions, unquestionedopinions, sanctioned customs and rites that do not call for justification.The transmission of this level of culture is based on direct admonition,explicit correction of errors without explanation. … Built up over gener-ations, the formal systems are normally very coherent. …

The ‘informal’ level is the level of the often unarticulated schemes ofbehaviour and thinking. Our knowledge of typing, or skiing, or bikingbelongs to this level of culture if we do not happen to be instructors ofthese skills. This level of culture is acquired through imitation, practiceand participation in a culture, and not by following a set of instructions.Very often neither the imitated nor the imitating know that someteaching–learning process is taking place.

At the ‘technical level’, knowledge is explicitly formulated. Thisknowledge is analytical, aimed to be logically coherent and rationallyjustified.

… technical education starts with errors and correction of errors, but adifferent tone is used here and the student is being explained his error(Hall, 1981).


The three levels are the basis for a description of mathematical modelling asa process (see p. 190).

One person’s useful distinctions (for example, ‘three worlds’) is anotherperson’s artificial and unhelpful separation. Some stances are based aroundachieving unity rather than distinction, but then make their own distinctionsin other ways. For instance, an enactivist stance (see Maturana, p. 70), asocial interactionist stance, and a social constructivist stance might questionthe notion of worlds, claiming that there is only the world of experience ofthe individual, or the world of social interactions!

Learning is not a single event but a continual adjustment (assimilation–accommodation) to impacts from the environment (physical and social,


imagined, symbolic), and learning is partly about gaining flexibility inmoving between these three worlds in which we all participate.

Learning as social interaction

Learning as participation in practices: Jean Lave and Etienne Wenger

Lave is a social anthropologist interested in learning as a social practice. Shewas inspired by the findings of Terezinha Nunes and colleagues (p. 17 and p.78), and discovered immense differences between people’s competence inarithmetic when in a practical situation such as outside a supermarket, and inan educational setting. This prompted her to consider traditional forms oflearning such as apprenticeships that still function in the Middle and Far East.Etienne Wenger (1952–) was a student and is now a colleague of Jean Lave.They teach that:

… all learning is situated in practice and represents a progression fromlegitimate peripheral participation to more central, expert participationin that practice; learning can be seen as a form of apprenticeship.

(adapted from Lave and Wenger, 1991)

Learning as social: Etienne Wenger

Wenger collaborated with Lave in the development of an influential socialperspective on learning.

Our institutions are largely based on the mistaken assumption thatlearning is an individual process, that it is best separated from the rest ofour activities, and that teaching is required for learning to occur.

(Wenger in Teplow, webref)

Wenger offers seven principles of learning:

1 Learning is inherent in human nature. Learning happens all the timeregardless of intention, design, or expectation. What is learned maynot reflect what is taught nor is it necessarily good for us or ourorganization.

2 Learning is fundamentally social. The opportunity for social partici-pation drives learning and makes it meaningful. Our thoughts,works, concepts, images and symbols reflect our social participation.Therefore, learning is most effective when it occurs within thecontext of social participation.

3 Learning changes who we are because it transforms our relationshipswith the world and our identities as social beings.


4 Learning is a matter of engagement in practice. Learning enables usto engage in the world in certain ways, participate in socially-definedactivities and contribute to a community and its enterprise. Theengagement in practice determines what we learn and our ability tocontribute to the community.

5 Learning reflects participation in communities of practice. Sharedenterprise promotes the development of a common practice, sharedways of doing things and relating to each other that enable individ-uals to achieve shared and individual objectives. Over time, recog-nizable bonds are forged among members who share a commonpractice and a community of practice forms. Communities of prac-tice are infinitely varied: formal or informal, enduring over centuriesor over the course of a single project, productive and harmonious ordestructive and adversarial. Learning is both the vehicle and theresult of participation in a community of practice. Finally, communi-ties of practice provide deep knowledge established over time andwith the potential to create new knowledge.

6 Learning means dealing with boundaries. Communities of prac-tice, by definition, create boundaries between participants andnon-participants. Learning is the recognition and reconciliation ofthese boundaries. In successful learning organizations, boundariesof practice can be leveraged by promoting interaction andinnovation.

7 Learning is an interplay between the local and the global. Multiplecommunities of practice usually exist within an organization, creatingconstellations of interrelated communities of practice. Local commu-nities of practice are the locus of work that reflects and affectsbroader organizational issues and relationships. Whereas the rela-tionship between an individual and the organization may be obscure,participation in local communities of practice provides the link. It iswhere learning takes place and where the meaning of belonging tothe organization is forged and experienced.

(ibid.)

Interactionism: Heinrich Bauersfeld

Heinrich Bauersfeld (b. Germany, 1926–) is a philosophical mathematicseducator with deep concern for the social interactions that form the core ofteaching and learning. Here he contrasts his stance with individualistic andcollectivist perspectives:

Individualistic perspectives: learning is the individual change, accordingto steps of cognitive development and to context. Prototype: cogni-tive psychology.


Collectivist perspectives: learning is enculturation into pre-existing soci-etal structures, supported by mediator means or adequate representa-tions. Prototype: activity theory.

Interactionist perspectives: teacher and students interactively constitutethe culture of the classroom, conventions both for subject matter andsocial regulations emerge, communication lives from negotiation andtaken-as-shared meanings. Prototypes: ethnomethodology, symbolicinteractionism, discourse analysis.

(Bauersfeld, 1993, p. 137)

[The core convictions of our interactionist position are, in brief, asfollows:]

1 Learning is a process of personal life formating, a process of an inter-active adapting to a culture through active participation (which inparallel also produces and develops the culture itself) rather than atransmission of norms, knowledge, and objectified items.

2 Meaning lies with the use of words, sentences, or signs and symbolsrather than in the related sounds, signs, or pictures …

3 Languaging … is a social practice, serving in communication forpointing at shared experiences and for orientation in the sameculture, rather than an instrument for the direct transportation ofsense or as a carrier of attached meanings.

4 Knowing or remembering something denotes an actual activation ofoptions from experienced actions [in their totality] rather than astorable, [deliberately] treatable, and retrievable object-like item,called knowledge, from a loft, called memory.

5 Mathematising is a practice based on social conventions rather thanthe applying of a universally applicable set of the eternal truths … .

6 (Internal) Representations are individual constructs, emergingthrough social interaction as a viable balance between the person’sactual interests and her realised constraints, rather than an internalone-to-one mapping of something pre-given or a fitting reconstruc-tion of ‘the’ world.

7 Using visualisations and embodiments with the related intention ofusing them as didactical means depends on taken-as-shared socialconventions [ … ] rather than on a plain reading or the discovering ofinherent or inbuilt mathematical structures and meanings.

8 Teaching describes the attempt to organise an interactive and reflec-tive process, with the teacher engaging in a constantly continuingand mutual[ly] differentiating and actualising of activities with thestudents, and thus the establishing and maintaining of a classroom‘culture’, rather than the transmission, introduction, or even re-discovery of pre-given and objectively codified knowledge.

(Bauersfeld, 1992, pp. 20–1)


Learning as engaging in cultural practices: Terezinha Nunes

Terezinha Nunes (b. Brazil, 1947–) is a psychologist who concentrates onmathematical learning. She studied street children in Brazil, before movingto England. She is well known for designing clever experiments to test theo-ries. Her research interests include the learning of number by deaf children.Here she summarises an article in which she takes the view that mathematicsis a cultural practice.

This article offers an integrated theoretical perspective of work wheremathematics is defined as a cultural practice. The implication of this defi-nition is that to learn mathematics is to become socialized into the waysof knowing used in the community of mathematicians and mathematicsteachers. Four aspects of the process of socialization are discussed: theredescription of meanings to fit with the systems of signs learned inmathematics, the influence of the connections created by teachers in theclassroom between the new concepts and the old meanings, the conse-quences of using particular systems of signs as mediators in reasoning,and the development of social representations of what mathematics is(and the associated process of valorization of particular methods) thattakes place in the classroom. Implications for multicultural classroomsare briefly discussed.

(Nunes, 1999, p. 33)


Learning has been described as being about making distinctions, restructuringattention, educating awareness, developing intuitions, responding to distur-bance, taking action and engaging in social interaction. Effective learning ofmathematics depends on, even lies in, the nature of social interactions. Effec-tive teaching draws on prior experiences rather than trying to displace them.

3 Analysis of learning forinforming teaching Analysis of learning for informing teaching

Introduction

Implications for teaching of the theories put forward in the previous chapterare many and varied depending on which aspects are stressed and which arerelatively ignored. In this chapter the influential approach to learning and todesigning teaching led by Guy Brousseau in France is developed andexplored. Conditions for effective learning are very complex and inter-twined. In addition to the sorts of assumptions and theories described in theprevious chapter there are more detailed tensions and issues to be examinedand closely probed.

Theory of didactic situations

Guy Brousseau (b. Morocco, 1933) is a leading researcher and theorist inFrance. He has built a substantial theory and practice based on observations,on some earlier constructs such as epistemological obstacles (p. 303) andtransformation didactique (p. 83), making use of Vygotsky’s stress on thesocial aspects of classrooms (see p. 86). The theory leads to systematicresearch into and development of teaching materials, in a process calleddidactic engineering (see Laborde, 1989; Artigue, 1993).

Didactic contract: Guy Brousseau and Michel Otte

The situation didactique identified by Guy Brousseau consists of thelearners, the teacher, the mathematical content and the classroom ethos, aswell as the social and institutional forces acting upon that situation, includinggovernment directives such as a National Curriculum statement, inspectionand testing regimes, parental and community pressures and so on. Withinthe situation didactique, Brousseau identified an implicit contract (contratdidactique) between teacher and learners, together with some concomitantforces, pressures and tensions.

The didactic contract is that ‘the teacher is obliged to teach and the pupilto learn’ (Brousseau and Otte, 1991, p. 18), or at least to pass the assessment.

The teacher sets tasks, the learners carry them out; the contract is that bydoing the tasks the learners will do enough to pass.

The contract must be honored at all costs, for otherwise there will be noeducation. Yet to be obeyed, the contract must be broken, becauseknowledge cannot be transmitted ready-made and hence nobody –neither the teacher nor the pupil – can be really in command.

(Brousseau and Otte, 1991, p. 180)

Situation didactique: Colette Laborde

Colette Laborde (b. France, 1946–) is a leading French educator andresearcher. Here she traces the origins of a collective stance towards educa-tional research.

… One of the concerns widely shared within the French community isthat of setting up an original theoretical framework developing its ownconcepts and methods and satisfying three criteria: relevance in relationto observable phenomena, exhaustivity in relation to all relevantphenomena, consistency of the concepts developed within the theoret-ical framework (Brousseau, 1986). …

What we call didactique des mathématiques in France covers thestudy of the relationships between teaching and learning in thoseaspects which are specific to mathematics. One widespread ideologypresupposes a connection of simple transfer from teaching towardslearning: the pupil records what is communicated by teaching withperhaps some loss of information. Numerous studies conducted withinPME [an annual international conference] have shown clearly howmistaken this point of view is by highlighting in particular the character-istics of the concepts constructed by pupils concerning arithmetic, alge-braic or geometric notions which are not contained in the teachingdiscourse: these concepts are local, partial and even false. These obser-vations forewarn us of the complexity of the links between teaching andlearning. This complexity is the origin and at the core of our research.

[…]French research in didactics has shown a desire to apprehend

teaching situations globally, to develop a modelization which encom-passes their epistemological, social, and cognitive dimensions andwhich attempts to take into account the complexity of the interactionsbetween knowledge, pupils and teachers within the context of a partic-ular class, or more generally of an educational group. …

One of the axes of research in didactics consists in extracting theconstraints which influence the didactic system and analyzing how theyfunction. The most important of these constraints are:


1 the characteristics of the knowledge to be taught, in particular thedependence between the mathematical objects which must be takeninto account in the creation of the coherence in the content to betaught;

2 the social and cultural constraints which act within the educationalproject to determine the teaching content;

3 the temporal characteristics of teaching fixed by the syllabi, in partic-ular its linearity;

4 the pupils’ concepts, their modes of cognitive development whichcondition access to new knowledge;

5 the teacher–learner asymmetry in relation to the knowledgeembedded in teaching situations (didactic contract) [see p. 79];

6 the teachers’ knowledge, their ideas and beliefs about mathematics,teaching, learning, and their own profession.

These constraints act together and have only been separated in order tobe exposed. They do not all occur at the same levels of the teachingprocess; constraints 1, 2, 3 and 4 particularly affect the determination ofthe knowledge to be taught (didactic transposition [see p. 83]) in theupper part of the teaching process, whereas constraints 4, 5, and 6 operatemore especially in the lower part, where the teaching is carried out.

[…]Three types of choice are fundamental, those relative to

• the choice of content to be taught;• the planning of interactions between learners and the knowledge to

be learnt;• the interventions and role of the teacher in a class situation.

The choice of teaching content and its organization is based onepistemological-type hypotheses and learning hypotheses… .

In the practice of teaching there is generally a process ofcontextualization of the knowledge taught – that is to say, the organiza-tion of a context which situates this knowledge, in which the activity ofthe pupils can operate. The interactions between knowledge and pupilsoperate through the context, the milieu (Brousseau, 1986). The antici-pated interactions can arise from different choices.

(Laborde, 1989, pp. 31–3)

See also using situations (p. 249).Of central importance are the questions to which the learners actually

reply (not necessarily the one asked by the teacher) and the ways in whichthe situation itself offers learners opportunities to check and modify theirthinking. Consequently, problems (see p. 45 and p. 303) play a central role inthe research, with learners expected to be active (see p. 53).

Analysis of learning for informing teaching 81

Didactic tension

Next comes a paradox of learner–teacher interaction and task setting:

… the ‘paradox of the didactic contract’ between teacher and learner. Ifboth the problem and the information about its solution are communi-cated by the teacher this deprives the pupil of the conditions necessaryfor learning and understanding. The pupil will only be able to reproducethe method of handling and solving the problem communicated to her.… mathematics is not just a method. We do know, on the other hand,that isolated problem situations do not of themselves produce means fortheir analysis or solution. And, in addition to that: what would be therelevance of the specific application of knowledge or of the solving of aparticular problem?

(Brousseau and Otte, 1991, p. 121)

The paradox, inescapably present in any classroom or teaching situation, canusefully be thought of as an endemic didactic tension:

Everything he does to make the pupil produce the behaviour he expectstends to deprive the pupil of the conditions necessary for understandingand learning the notion concerned. If the teacher says what he wants, hecan no longer obtain it.

(Brousseau, 1984, p. 110)

… most of the time the pupil does not act as a theoretician, but as a prac-tical man. His job is to give a solution to the problem the teacher hasgiven to him, a solution that will be acceptable in the classroom situa-tion. In such a context the most important thing is to be effective. Theproblem of a practical man is to be efficient, not to be rigorous. It is toproduce a solution, not to produce knowledge.

(Balacheff, 1986, quoted in Sierpinska, 1994, p. 19)

The more explicit I am about the behaviour I wish my pupils todisplay, the more likely it is that they will display that behaviourwithout recourse to the understanding which the behaviour is meant toindicate; that is, the more they will take the form for the substance.

The less explicit I am about my aims and expectations about thebehaviour I wish my pupils to display, the less likely they are to noticewhat is (or might be) going on, the less likely they are to see the point,to encounter what was intended, or to realise what it was all about.

(Mason, 1988, p. 33)

The didactic tension creates a dilemma for teachers. If you think that learnersultimately have to reconstruct ideas for themselves, how do you arrange for


learners to reconstruct accurately and appropriately without ‘telling them’? Ifyou think that it is through being immersed in practices that learners adoptthe language and habits of experts and so learn to become relatively expertthemselves, then what sorts of practices are effective, and what has tohappen during the ‘immersion’? If you think that it is possible to preparelearners to be in a position to hear what the teacher-expert has to say, whatadvice they have to give, then although it seems possible to achieve learningthrough direct instruction, what do you do when learners misconstrue, orconstruct meanings and approaches and ways of thinking different to thoseof the teacher?

Problématique: Nicolas Balacheff

Nicolas Balacheff (b. Germany, 1947–) is a leading researcher in mathematicseducation in France. He helped to expose to the English speaking researchcommunity various constructs and approaches coming from colleagues suchas Guy Brousseau (situation didactique, see p. 79), Yves Chevellard(tranposition didactique the expert awareness is transposed into instructionin behaviour) and Gaston Bachelard (obstacles, see p. 303). Balacheff alsopioneered a constructivist perspective, both psychological and social, in theanalysis of classroom teaching.

A problématique is a set of research questions related to a specifictheoretical framework. It refers to the criteria we use to assert that theseresearch questions are to be considered and to the way we formulatethem. It is not sufficient that the subject matter being studied is mathe-matics for one to assert that such a study is research on mathematicsteaching. A problem belongs to a problématique of research on mathe-matics teaching if it is specifically related to the mathematical meaning ofpupils’ behavior in the mathematics classroom. …

Our theoretical framework is grounded on two hypotheses: theconstructivist hypothesis and the epistemological hypothesis.

The constructivist hypothesis is that pupils construct their own knowl-edge, their own meaning. The fact that previous knowledge is ques-tioned, the disequilibration in the Piagetian sense, results in theconstruction of new knowledge as a necessary response to the pupils’environment.

The epistemological hypothesis (Vergnaud, 1982) is that problems arethe source of the meaning of mathematical knowledge, but also intellec-tual productions turn into knowledge only if they prove to be efficientand reliable in solving problems that have been identified as beingimportant practically (they need to be solved frequently and thuseconomically) or theoretically (their solution allows a new under-standing of the related conceptual domain).

(Balacheff, 1990, pp. 258–9)


There are similarities with disturbance (see p. 55) and surprise (see p. 101).

We consider the aim of teaching to be carrying pupils from their initialconceptions related to a given item of mathematical knowledge to resul-tant conceptions through what we call a didactical process. The controland design of this didactical process constitutes the heart of our approach.

… the fundamental means to initiate this process are mathematicalproblems. Mathematical problems are fundamental insofar as theyconstitute means to challenge the pupils’ initial conceptions and toinitiate their evolution. Also, they are fundamental because they conveythe meaning of the mathematical content to be taught mainly by makingexplicit the epistemological obstacles [see p. 303] that must be overcomefor the construction of that meaning.

Pupils’ behaviors in the context of a classroom situation cannot beunderstood only through an analysis of the mathematical contentinvolved or its related psychological complexity. The problems offeredto pupils in a didactical situation are set in a social context dominated byboth explicit and implicit rules that permit it to function but also that givemeaning to pupils’ behaviors. … The rules of social interaction in themathematics classroom include such issues as the legitimacy of theproblem, its connection with the current classroom activity, and respon-sibilities of both the teacher and pupils with respect to what constitutes asolution or to what is true. We call this set of rules a didactical contract. Arule belongs to the set, if it plays a role in the pupils’ understanding of therelated problem and thus in the constitution of the knowledge theyconstruct.

(ibid., p. 260)

A useful history and background for French didactiques can be found inLaborde (1989) and in Caillot (2002). A similar Italian approach is describedin Bussi (webref).

Activity theory

Lev Vygotsky (b. Belarus, 1896–1934) was a psychologist who developed anapproach that has come to be known as activity theory. Vygotsky’s ideaswere brought to the west by Jerome Bruner and by James Wertsch (1991)


In designing tasks, it is important to bear in mind the didactic transposition byconstructing tasks which provoke the learner to reconstruct distinctions, rela-tionships, properties, etc..

Over time, the force is for learners to work out what behaviour is requiredand to produce that, while the teacher is looking for that behaviour to begenerated or constructed by the learners (didactic tension).

and developed by many mathematics teacher-educators and reseachers,including Steve Lerman from whose writing the following extract is taken.

Activity theory: Lev Vygotsky

Lerman extracts from Vygotsky’s work four key elements as a mechanism forlearning: the priority of the inter-subjective; internalisation; mediation; andthe zone of proximal development:

• Every function in the child’s cultural development appears twice: first,on the social level, and later, on the individual level; first betweenpeople (interpsychological), and then inside (intrapsychological). …All the higher functions originate as actual relations between humanindividuals.

(Vygotsky, 1978, p. 57, quoted in Lerman, 2000, p. 34)

• …the process of internalization is not the transferral of an external[activity] to a pre-existing, internal ‘plane of consciousness’; it is theprocess in which this internal plane is formed.

(Leont’ev, 1979, p. 57, quoted in Lerman, 2000, p. 34)

• Human action typically employs ‘mediational means’ such as toolsand language, and that these mediational means shape the actions inessential ways … the relationship between action and mediationalmeans is so fundamental that it is more appropriate, when referring tothe agent involved, to speak of ‘individual(s) acting with mediationalmeans’ than to speak simply of ‘individual(s)’.

(Wertsch, 1991, p. 12, quoted in Lerman, 2000, p. 34)

• We propose that an essential feature of learning is that it creates thezone of proximal development; that is, learning awakens a variety ofdevelopmental processes that are able to interact only when the childis interacting with people in his environment and in cooperation withhis peers.

(Vygotsky, 1978, p. 90, quoted in Lerman, 2000, p. 34)

Activity theory: Aleksej Leont’ev

Leont’ev was a colleague of Vygotsky and principal theorist of activity theory.Wertsch (1991, p. 255, pp. 86–7, and pp. 264–5), describes Leont’ev asdefining three levels of activity: energised or motivated activity, action definedby a goal, and operational. Operational activity is unaware of goals orsubgoals, and energised or motivated activity may not have conscious goals.

For Leont’ev, sense designates personal intent, as opposed to meaning,which is public, explicit, and literal. Sense derives from the relations of


actions and goals to motivated (higher order) activities of which they are aparticular realisation. Zinchenko showed that ‘material that is the immediategoal of an action is remembered concretely, accurately, more effectively,more durably. When related to the means of an action (to operations) thesame material is remembered in a generalized way, schematically, less effec-tively, and less durably’ (ibid., p. 317). Activity theory influenced those whodeveloped the notion of situated cognition (cognition is always limited bythe situation in which it occurs: see p. 292), and there are similarities withtheorem-in-action (see p. 63), which could be seen as undifferentiatedwholistic behaviour in which the actor is unaware of the principles by orthrough which they are acting.

In the next extract Leont’ev compares experiences of others with innerexperiences in order to support the view that everything is first experiencedsocially before being internalised.

… Vygotsky laid the foundations, in his early works, for analyzingactivity as a method of scientific psychology. He introduced theconcepts of the tool, tool (‘instrumental’) operations, the goal, and –later – the motive (‘the motivational sphere of consciousness’). …

[…]… we always deal with specific activities. Each of these activities

answers to a specific need of the active agent. It moves toward the objectof this need, and it terminates when it satisfies it. Also, it may be repro-duced under completely different circumstances. Various concrete activ-ities can be classified according to whatever features are convenient,such as form, means of execution, emotional level, temporal and spatialcharacteristics, physiological mechanisms, etc.. However, the mainfeature that distinguishes one activity from another is its object. After all,it is precisely an activity’s object that gives it a specific direction. … anactivity’s object is its real motive. …

(Leont’ev, 1979, p. 59)

There are similarities with purpose (see p. 42) and motivation (see p. 99).

The basic ‘components’ of various human activities are the actions thattranslate them into reality. We call a process an action when it is subor-dinated to the idea of achieving a result, i.e., a process that is subordi-nated to a conscious goal. Just as the notion of a motive is tied to anactivity, so the notion of a goal is connected with the notion of anaction. The emergence in activity of goal-directed processes or actionswas historically the consequence of the transition of humans to life insociety.

(ibid., pp. 59–60)


Higher and lower psychological processes: Lev Vygotsky andHeinrich Bauersfeld

Bauersfeld succinctly compares lower and higher psychological processes:

In a transient phase of this thinking about 1930, Vygotsky discriminatedhigher from lower mental functions through their genesis. The lowermental functions follow stimulus–response constellations; they developthrough maturation. In contrast, higher mental functions are mediatedthrough the use of tools and signs, and are open to conscious and delib-erate training. The higher functions develop only within societal rela-tions, ‘through the internalisation of self-regulatory pattern pre-given insociety’… .


An important feature of learning for Vygotsky was the role of relative expertsin displaying higher psychological processes to be picked up and internalisedby novices, during which they are generalised, verbalised, abbreviated andso afford the possibility of further development. Higher psychologicalprocesses include the use of tools (physical and virtual) including mostimportantly speech and signs, but also things like tables for laying out data,formats for doing arithmetical calculations, use of diagrams to depict anddisplay, and so on.

… higher psychological processes unique to humans can be acquiredonly through interaction with others, that is, through interpsychologicalprocesses that only later will begin to be carried out independently bythe individual. When this happens, some of these processes lose theirinitial, external form and are converted into intrapsychologicalprocesses.

… the process of interiorization is not the transferal of an externalactivity to a pre-existing, internal ‘plane of consciousness’: it is theprocess in which this internal plane is formed.

(Leont’ev, 1979, pp. 56–7)

… the general genetic law of cultural development… : Any function inthe child’s cultural development appears twice, or on two planes. First itappears on the social plane, and then on the psychological plane. First itappears between people as an interpsychological category, and thenwithin the child as an intrapsychologicial category. This is equally truewith regard to voluntary attention, logical memory, the formation ofconcepts, and the development of volition. … Social relations or rela-tions among people genetically underlie all higher functions and theirrelationships.

(Vygotsky, 1979, p. 163)


Zone of proximal development: Lev Vygotsky

Bruner used Vygotsky’s ideas as the basis of his research on the role of atutor (see p. 89). Here is Vygotsky:

… consciousness and control appear only at a late stage in the develop-ment of a function, after it has been used and practised unconsciouslyand spontaneously. In order to subject a function to intellectual …control, we must first possess it.

(Vygotsky, 1962, p. 90)

Compare this with ‘teaching for understanding’ (see p. 293) and educatingawareness (see p. 61 and p. 204).

Bruner translates the previous Vygotsky extract into more accessiblelanguage, and explains what Vygotsky meant by the zone of proximaldevelopment.

… prior to the development of self-directed, conscious control, action is,so to speak, a more direct or less mediated response to the world.Consciousness or reflection is a way of keeping mind from (if the mixedmetaphor will be permitted) shooting from the hip. That much is familiarenough as a form of conscious inhibition. But what of the instruments bymeans of which mind now grapples itself to ‘higher ground’?

This is the heart of the matter, the point at which Vygotsky brings tobear his fresh ideas about the now famous Zone of Proximal Develop-ment (the ZPD hereafter). It is an account of how the more competentassist the young and the less competent to reach that higher ground,ground from which to reflect more abstractly about the nature of things.To use his words, ‘the ZPD is the distance between the actual develop-mental level as determined by independent problem-solving and thelevel of potential development as determined through problem-solvingunder adult guidance or in collaboration with more capable peers.’(Vygotsky, 1978, p. 86). ‘Human learning’ he says, ‘presupposes aspecific social nature and a process by which children grow into theintellectual life of those around them’ (Vygotsky, 1978, p. 88). … ‘Thusthe notion of a zone of proximal development enables us to propound anew formula, namely that the only ‘good learning’ is that which is inadvance of development’ (Vygotsky, 1978, p. 89).

(quoted in Bruner, 1986, p. 73)

Consciousness for two: Jerome Bruner

Bruner comments that nowhere in Vygotsky’s writing could he find specificdescriptions of how a tutor could scaffold the learner’s experience in order


that consciousness and control can arise after the function is well and sponta-neously mastered.

The tutor, Dr Ross, was not only knowledgeable about children but genu-inely interested in what they were doing and how they could be helped.… she turned the task into play and caught it in a narrative that gave itcontinuity.

What emerged was, I suppose, obvious enough. She was indeed ‘con-sciousness for two’ for the three- and five-year-olds she tutored, and inmany ways. To begin with, it was she who controlled the focus of attention.It was she who, by slow and often dramatized presentation, demonstratedthe task to be possible. She was the one with the monopoly on foresight.She kept the segments of the task on which the child worked to a size andcomplexity appropriate to the child’s powers. She set things up in such away that the child could recognize a solution and perform it later eventhough the child could neither do it on his own nor follow the solutionwhen it was simply told to him. In this respect, she made capital out of the‘zone’ that exists between what people can recognize or comprehendwhen present before them, and what they can generate on their own – andthat is the Zone of Proximal Development, or the ZPD. In general, what thetutor did was what the child could not do. For the rest, she made thingssuch that the child could do with her what he plainly could not do withouther. And as the tutoring proceeded, the child took over from her parts ofthe task that he was not able to do at first but, with mastery, becameconsciously able to do under his own control. And she gladly handed thoseover. (Interestingly, when the observations were repeated years later usingyoung children as tutors for younger children, they were not as different asexpected, save in one crucial respect: the young tutors would not handover parts of the task as the younger child achieved mastery.)

(Bruner, 1986, pp. 75–6)

Note the importance for Bruner of narrative (see p. 68).

Realm of developmental possibilities: Paul Cobb

Paul Cobb (b. Hertfordshire, 1953–) moved to the USA as a researcher and hasled the shift to focus more strongly on the role of the social and cultural inlearning mathematics in classrooms as the basis for current thinking.

During the analysis of the interviews, frequent reference is made to theconceptual interpretations that the children seemed to make, and to thearithmetical objects they are inferred to have constructed. This way oftalking is a shorthand used for ease of explication. The intendedmeaning in each case is that, in the course of his or her participation inthe interview, the child acted in ways that justify making particular


cognitive attributions. The situated nature of these inferences is particu-larly apparent in those instances in which the researcher attempted tosupport the child’s mathematical activity. In several of these instances,children did in fact seem to make conceptual advances that they wouldnot, in all probability, have made on their own.

(Cobb, 1995, p. 29)

Note the care in not over-reaching what can be said about what learners areobserved doing.

One standard way of accounting for these advances is to use Vygotsky’sconstruct of the ‘zone of proximal development’. However, this notionelevates interpersonal social processes above intrapersonal cognitiveprocesses (Cole, 1985). Thus, analyses that use this construct typicallyfocus on the adult’s role in scaffolding the child’s activity. As a conse-quence, the treatment of both the child’s interpretations and his or hercontributions to interactions is relatively limited. The zone of proximaldevelopment was therefore replaced by a construct that is more relevantto the purposes of the investigation, that of the ‘realm of developmentalpossibilities’. This construct delineates the situated conceptual advancesthe child makes while participating in an interaction such as that in whichan adult intervenes to support his or her mathematical activity. The zoneof proximal development is concerned with what the child can do withadult support, whereas the realm of developmental possibilities addressesthe way in which the child’s conceptions and interpretations evolve as heor she interacts with the adult. The latter construct, therefore, brings thecognitive perspective more to the fore and, thus, complements sociolog-ical analysis of the situations in which that development occurs.

(ibid., p. 29)

Attention: Lev Vygotsky

Vygotsky notes the importance of learning to direct attention.

Attention should be given first place among the major functions in thepsychological structure underlying the use of tools. … scholars havenoted that the ability or inability to direct one’s attention is an essentialdeterminant of the success or failure of any practical operation.However, the difference between the practical intelligence of childrenand animals is that children are capable of reconstructing their percep-tion and thus freeing themselves from the given structure of the field.With the help of the indicative function of words, the child begins tomaster his attention, creating new structural centers in the perceivedsituation. … the child is able to determine for herself the ‘center of grav-ity’ of her perceptual field; her behavior is not regulated solely by the


salience of individual elements within it. The child evaluates the relativeimportance of these elements, singling out new ‘figures’ from the back-ground and thus widening the possibilities for controlling her activities.

In addition to reorganizing the visual–spatial field, the child, with thehelp of speech, creates a time field that is just as perceptible and real tohim as the visual one. The speaking child has the ability to direct hisattention in a dynamic way. He can view changes in his immediate situa-tion from the point of view of past activities, and he can act in thepresent from the viewpoint of the future.

(Vygotsky, 1978, pp. 35–6)

There are similarities with three worlds (see p. 73) and proximal relevance(see p. 110).

Labelling and distinctions: Lev Vygotsky

Note in this extract the role of labelling in assisting and preserving distinc-tions discerned.

A series of related observations revealed that labeling is the primary func-tion of speech used by young children. Labeling enables the child tochoose a specific object, to single it out from the entire situation he isperceiving. Simultaneously, however, the child embellishes his first wordswith very expressive gestures, which compensate for his difficulties incommunicating meaningfully through language. By means of words chil-dren single out separate elements, thereby overcoming the natural structureof the sensory field and forming new (artificially introduced and dynamic)structural centers. The child begins to perceive the world not only throughhis eyes but also through his speech. As a result, the immediacy of ‘natural’perception is supplanted by a complex mediated process; as such, speechbecomes an essential part of the child’s cognitive development.

[…]The role of language in perception is striking because of the opposing

tendencies implicit in the nature of visual perception and language. Theindependent elements in a visual field are simultaneously perceived; in thissense, visual perception is integral. Speech, on the other hand, requiressequential processing. Each element is separately labeled and thenconnected in a sentence structure, making speech essentially analytical.

Our research has shown that even at very early stages of develop-ment, language and perception are linked. In the solution of nonverbaltasks, even if a problem is solved without a sound been uttered,language plays a role in the outcome.

(Vygotsky, 1978, pp. 32–3)


Constructivisms

Piaget’s notion of genetic epistemology came to be referred to as constructivism,with roots in Vico (see p. 53) and other earlier writers. The version that wastaken up in education was rather simplified, so Ernst von Glasersfeld introducedradical constructivism to stress the less comfortable aspects. However, manyfelt that the individual was overemphasised at the expense of the social, sosocial constructivism redressed this, but in many cases was taken to an extremeas radical social constructivism. Meanwhile in the context of robots andcomputers the notion of constructionism emphasised physical construction ofobjects.

Constructivism: Jean Piaget

Jean Piaget (b. Switzerland, 1896–1980) was a biologist turned philospherconcerned with how organisms, especially humans, learn. He was head of aMontessori school before becoming a researcher. In collaboration with manycolleagues over the years, Piaget devised numerous probes for revealing thedevelopment of children’s thinking, and these investigations continue at thePiaget Institute in Geneva. His experiments and conclusions profoundlyinfluenced Western education.

Piaget formulated the notion of genetic epistemology to capture his senseof children being genetically prepared to construct knowledge of the worldfor themselves, with, of course, the support and under the influence oftheir social and cultural environment. This includes direct instruction theyreceive from teachers (including parents) as well as the myriad of implicitforces and practices in which they are imbedded. Primary among these islanguage, for it is through language that we express ourselves to others,and show that we can participate socially. Various forms of constructivismhave been proposed and advocated as ‘telling the best story’ about whatlearners do.

The various constructivisms are a response to the challenge to explainhow learners construct the mathematical ways of knowing as they interactwith others in the course of their mathematical acculturation (Cobb, Yackeland Wood, 1992, p. 17).


Learners need not just to be ‘active’, but to be in the presence of others whodisplay the ‘higher psychological functioning’ before they can be expected tointernalise it for themselves.

Labelling distinctions in language enables distinctions to be made in thefuture, and this applies to identifying relationships, properties and so on.

Radical constructivism: Ernst von Glasersfeld

The basic principle of the constructivist theory is that cognitive organ-isms act and operate in order to create and maintain their equilibrium inthe face of perturbations generated by conflicts or unexpected noveltiesarising either from their pursuit of goals in a constraining environment orfrom the incompatibility of conceptual structures with a more or lessestablished organization of experience. The urge to know thus becomesthe urge to fit; on the sensory-motor level as well as in the conceptualdomain, and learning and adaptation are seen as complementaryphenomena.

If one accepts this principle, one can no longer maintain the tradi-tional idea of knowledge as representing an ‘external’ reality supposedto be independent of the knower. The concept of knowledge has to bedismantled and reconstructed differently. This is a shocking suggestion,and I have elsewhere laid out the reasons for such a radical step (vonGlasersfeld, 1985). I have called my position radical constructivism toaccentuate the changed concept of knowledge and to differentiatemyself from those who speak of the construction of knowledge in theframework of a traditional epistemology. … It is intended to be used as aworking hypothesis whose value can lie only in its usefulness.


1 Training aims at the ability to repeat the performance of a givenactivity and it must be distinguished from teaching. What we want tocall teaching aims at enabling students to generate activities out ofthe understanding why they should be performed and, ultimately,also how one can explain that they lead to the desired result.

2 Knowledge has to be built up by each individual learner, it cannot bepackaged and transferred from one person to another.

3 Language is not a conveyor belt or means of transport. The meaningof words, sentences and texts is always a subjective constructionbased on the individual’s experience. Though language cannot ‘con-vey’ the desired constructs to students, it has two important func-tions: it enables the teacher to orient the students’ conceptualconstruction by means of appropriate constraints; and when studentstalk to the teacher or among themselves in groups, they are forced toreflect upon what they are thinking and doing.

4 Students’ answers and their solutions of problems should always betaken seriously. At the moment they are produced, they mostly makesense to the student even if they are wrong from the teacher’s pointof view. Ask students how they arrived at their answer. This helps toseparate answers given to please the teacher from those that are theresults of understanding or misunderstandings.


5 Only a problem the student sees as his or her own problem can focusthe student’s attention and energy on a genuine search for a solution.

6 Rewards (i.e., the behaviorists’ external reinforcements), be theymaterial or social, foster repetition, not understanding.

(ibid., pp. 25–6)

There are similarities with Spencer’s observations (see p. 35).

7 Intellectual motivation is generated by overcoming an obstacle, by elim-inating a contradiction, or by developing principles that are bothabstract and applicable. Only if students have themselves built up aconceptual model that provides an explanation of a problematic situa-tion or process, can they develop the desire to try their hand at furtherproblems; only success in these attempts can make them aware of theirpower to shape the world of their experience in a meaningful way.

(ibid., p. 26)

Note the use of learners’ powers (see p. 115 and p. 233).

Radical constructivism, thus, is radical because it breaks with conven-tion and develops a theory of knowledge in which knowledge does notreflect an ‘objective’ ontological reality, but exclusively an ordering andorganization of a world constituted by our experience. The radicalconstructivist has relinquished ‘metaphysical realism’ once and for alland finds himself in full agreement with Piaget, who says: ‘Intelligenceorganizes the world by organizing itself’.

(von Glasersfeld, 1984, p. 24)

Constructionism: Seymour Papert

Seymour Papert (b. South Africa, 1928–) started working life as a mathemati-cian. He worked with Piaget before moving to the USA where he became aleader in the development of the field of artificial intelligence. In his ground-breaking book (Papert, 1980), which promulgated the use of LOGO as acomputer program accessible to children, he maintained that learning is mosteffective when learners are constructing things through actions they perform,although his focus is largely on computers and computer-driven robots.

Constructionism is built on the assumption that children will do best byfinding (‘fishing’) for themselves the specific knowledge they need.Organized or informal education can help most by making sure they aresupported morally, psychologically, materially, and intellectually in theirefforts. … the goal is to teach in such a way as to produce the mostlearning for the least teaching.

(Papert, 1993, p. 139)


Social constructivism

In addition to the classic triad of subject matter, learner and teacher, there is theenvironment within which all the interactions take place, the milieu asBrousseau calls it (p. 91). Stressing the milieu is one way of describing acomplex and multifaceted perspective often referred to as social constructivism.An early advocate was Kurt Lewin who coined a number of pithy sayings thatare still much quoted today such as ‘If you want truly to understand something,try to change it’ (see also p. 161), but who is also famous for framing behaviouras both individual and social: ‘behaviour is a function both of the person and ofthe environment’. In the twenty-first century this seems self-evident to usperhaps, but it was an attempt to reconcile two competing strains ofpsychology, which continue to vie with each other for ascendancy. Stressing theperson leads to emphasis on psychology, and often specifically on cognition, onthinking, and on behaviour. Stressing the environment leads to emphasis on thesocial and on the role of language in its most general sense as the medium inwhich, and by means of which, we take up and use social practices of variouskinds, and also through which we are used by our culture. Stressing individualsas sense-makers leads to psychological constructivism, whereas stressing thesocial leads to social constructivism and stressing the individual–environmentpair leads to enactivism (p. 70).

Community of practice: Jean Lave

Lave and Wenger developed the notion of a community of practice in whichpeople experience the behaviour of relative experts.

A practice is any behaviour, including mental actions, which are typical ofa group of people (the community). As a member of a school there arecertain things you do, ways you dress, things you say; as a participant in alesson about mathematics there are typical things you learn to say and do.These are all practices. Practices are picked up from others in thecommunity.

This conforms with Vygotsky’s notion that what he called higher psycholog-ical processes are first experienced in the behaviour of others, and only laterinternalised into learners’ practices. As these ideas developed some peopleput forward the ‘radical’ notion that knowledge is not something possessed oraccessed by individuals, but rather resides in the social practices of differentcommunities. A simple social perspective holds that discussion and peripheralparticipation is a valuable component of learning; a radical perspective holdsthat learning consists in becoming adept at carrying out the practices(speaking and acting in recognised and established ways).

… a decentred view of the locus and meaning of learning, in whichlearning is recognized as a social phenomenon constituted in the experi-enced, lived-in world, through legitimate peripheral participation in


ongoing social practice; the process of changing knowledgeable skill issubsumed in processes of changing identity in and through membershipin a community of practitioners; and mastery is an organizational, rela-tional characteristic of communities of practice.

(Lave, 1993a, p. 64, also quoted in Boaler, 2000, p. 5)

Constructivist positions summarised

For an admirable summary of various constructivist positions linked togetherfrom a single perspective see (Confrey, 1990; 1991; 1994; 1994a; 1994b;1995). A useful source of writings on various forms of constructivism can befound at Selden and Selden (webref), in Steffe and Gale (1995) and on math-ematics education, specifically in Steffe et al. (1996).

A teaching dilemma: Paul Cobb

Cobb drew attention to an ever-present and apparently inescapabledilemma. As a teacher convinced that learners have to make sense, have toconstruct and reconstruct meaning for themselves, how is it possible for ateacher to arrange that learners construct what experts have constructed andnot something that is fallacious or confused? Is it possible to ensure that theyconstruct appropriate meanings and interpretations, construct useful modelsand ways of perceiving, notice what experts consider worthy of notice?

Furthermore, on the whole, meanings do seem to be shared, or at least, asPaul Cobb, Erna Yackel, and Teri Wood (1992, p. 8) put it, are ‘taken asshared’, perfectly effectively for the most part. Yet if learners are makingsense, are reconstructing meaning, are interpreting for themselves, how canthe considerable agreement that is achieved be accounted for?

Cobb looked initially for a resolution of this apparent paradox by placingconfidence in learners to adjust and accommodate over the long term.

Our approach is … that activities are provided for the teachers whoimplement them very much in a problem centred way. They are underno obligation to ensure that all children come up with a certain idea on aparticular day. The meanings emerge over a long period of time in thecourse of discussions. … The actual specifics of how do I get all thesechildren to see the relationship between these numbers does not arisefor our teachers.

(Cobb, Open University interview, July 1988)

Dissatisfied with this, he went on to stress social aspects, and to go so far asto place knowledge in the language and practices of the community ratherthan in the individual. Thus the learner adopts practices and ways ofspeaking. But this may overlook the experience that when something isbeing articulated, being expressed in some way (through diagrams and


pictures, through movement, sculpting and so on, and through words andsymbols), there is something of or about the individual that is beingexpressed and that is not entirely social.

Instead of going to an extreme social-constructivist position, it is tenable toremain with the observation that it is in the nature of evolving responsiveorganisms to adjust to the environment, whether in the sense of fitting usedby von Glasersfeld (see p. 93), through assimilating or accommodating(Piaget, p. 149) or in not seeing any distinction between the organism and itsenvironment, as in enactivism (see p. 70).

The dilemma is not confined to a constructivist perspective. It faces everyteacher no matter what their perspective, for learners will make sense inwhatever way they find they can. Telling learners something is sometimeseffective, though often not very effective (see p. 73, p. 116 and p. 229). Buteven where learners are told, if they are going to do more than recite it backagain from memory, if they are to integrate it into their awareness, then somechange, some transformation, some alteration in how they perceive, whatthey notice and how they act, and how they account to themselves for whatthey perceive and notice comes about.

Cobb’s response is that when tasks are sufficiently carefully designed, withattention paid to various principles described in these pages, and when suit-able conditions are established, learners will in fact come to similar conclu-sions about mathematical facts and theorems.

There is a mathematical version of the surprising agreement in meaning,which has exercised mathematicians: how can one account for the ‘unrea-sonable effectiveness of mathematics’ (Wigner, 1960 webref) for modelling,predicting, and controlling the material world, if mathematics is a construc-tion of the human mind?

See Hamming (1980; webref) for one reply, and Newman (1956) forothers.

Teaching dilemma: Derek Edwards and Neil Mercer

Derek Edwards (b. Liverpool, 1948–) and Neil Mercer (b. Northamptonshire,1948–) together tackled the dilemma in the context of teaching generally,with some special attention to science and mathematics. They were shockedto find little discussion and social negotiation of meaning in the classroomsthey observed.

… The teacher’s dilemma is to have to inculcate knowledge whileapparently eliciting it. This gives rise to a general ground-rule of class-room discourse, in which the pupils’ task is to come up with the correctsolutions to problems seemingly spontaneously, while all the time tryingto discern in the teacher’s clues, cues, questions and presuppositionswhat that required solution actually is.

(Edwards and Mercer, 1987, p. 126)


These tensions cannot be resolved. Being endemic means that energy isstored in them. Instead of trying to avoid the tension, the fact of the tensioncan serve as a constant reminder that learning and hence teaching aredynamically varying processes, which are never completed and whichrapidly change their nature over short and long periods of time. Attentiontherefore usefully moves to considering the actions that learners can under-take in order to learn, and from that, specific suggestions arise for teaching.


Ultimately, it is learners who make sense of the world through reconstructingideas for themselves (psychological constructivism). It is when learners areactively involved, and ideally literally as well as metaphorically building some-thing, that learning is most efficient (constructionism). The practices, especiallyuse of language and ways of behaving picked up from others, are for some theprincipal source from which meaning is constructed (social constructivism)and, for some, knowledge resides in the community and not in the individualas such (strong social constructivism).

4 Affect in learningmathematics Affect in learning mathematics

Introduction

Many factors affect what a learner learns from an educational activity. Themotivations of learners and the intentions of teachers inevitably differ. Otherfactors relate to the context within which activity takes place and the authen-ticity of the activity as experienced by the learner. When teachers, influencedby their own desires, interact with learners, they may influence the learner’smotivation and self-esteem. These issues are the subject of this chapter.

Motivation, intention and desire

Motivation is a complex matter. It is a word used to describe, literally, whatmoves learners. Teachers, and the school ethos and setting, play a significantrole, both in motivating and in demotivating. Both of these can be throughsubtle, covert features of ways of working and interacting with learners, andovertly through the provision of examples and contexts to which learnerscan readily relate.

Philosophers have long felt that mathematics provides an ideal context inwhich to encounter pleasure in the use of human powers. For example, Platoextols pure arithmetic:

… they must carry on the study until they see the nature of numbers inthe mind only …

[ … ]… pursued in the spirit of a philosopher and not of the shopkeeper![ … ]… arithmetic has a very great and elevating effect, compelling the soul

to reason about abstract number, and if visible or tangible objects areobtruding upon the argument, refusing to be satisfied.

(Plato: The Republic VII, p. 525, Jowett, 1871, p. 360)

Not all learners aspire to such dizzy heights! The presence of unmotivatedand unappreciative learners is no new phenomenon. The Hadow Report of

1931 found the need to ‘emphasise the principle that no good can come fromteaching children things that have no immediate use for them, howeverhighly their potential or prospective value may be estimated.’ It stated thatthe curriculum should be thought of not as knowledge and facts but asactivity and experience (Hadow Report, 1931, p. 73; quoted in McIntosh,1977, p. 94; reprinted in Floyd, 1981, p. 9.)

Motivating: Lancelot Hogben

Lancelot Hogben (b. Hampshire, 1895–1975) was an eminent zoologist whobecame an expositor of science and mathematics ‘for the millions’. Just likeRobert Recorde (b. Wales, 1510–1558) who wrote the first arithmetic andalgebra books in English, Hogben wanted to provide everyone with accessto the important and powerful ideas of science and mathematics. In a talk hegave in England in 1938 he formulated conclusions about the role of themathematics teacher. There are similarities with psychologising the subjectmatter (see p. 45 and p. 203) and use of learners’ powers (p. 115 and 233).

The primary task of an educationalist is to establish a personal relation-ship by enlisting the personal interest of individual pupils in the exerciseof their reasoning powers. Thus the problem of the mathematics teacheris not a problem of mathematics as such. The recipe for good mathe-matics teaching is to put into the teaching of mathematics somethingwhich does not belong to the subject-matter of mathematics as such.There are obviously many levels at which this can be done. … Theaesthetic, which, at its most primitive level, is the play motive, is onewhich an enthusiastic and efficient teacher will not neglect, though I donot think it carries us very far by itself.

[ … ]… let me urge that we should resist the temptation to make the exami-

nation system an excuse for lack of enterprise in education.… I believe that the teacher who … takes the trouble to stimulate his

pupils by devoting a substantial part of his time to topics which lie quiteoutside the syllabus will get better examination results than the teacherwho keeps one eye glued on the syllabus.

(Hogben, 1938, pp. 111–3)

Motivation: Richard Skemp

Richard Skemp (b. Avon, 1919–1995) took both a mathematics degree andthen a psychology degree, and pioneered the use of psychology withinmathematics education in the UK. In addition to curriculum development, hedeveloped a sophisticated model of intelligence, which integrates cognitiveand emotional dimensions. One component is the notion of goal-states (tobe sought) and anti-goal-states (to be avoided). One person may wish to


avoid challenge (see Dweck, p. 112) while another relishes it; one may avoidconflict with teachers, and another may seek it out as a means of gettingattention from adults and/or from peers. Skemp points out (1979, p. 15) thatwhere there is no novelty in a situation, there is likely to be a low level ofconsciousness; where there is excessive novelty, the learner may not be ableto cope and so blocks off inputs; where there is some novelty but not exces-sive amounts, consciousness is likely to be at a maximum and so the learneris in the best position to learn.

Can one person motivate another?[ … ]The general sense in which ‘A motivates B’ is used means, roughly,

that A gets B to do something that A wants B to do, and which B wouldnot otherwise have done. With this is an implication that the action by Ais intentional, and unilateral. …

What can A do whereby to bring about the action which he wants B todo? A can command or threaten, strongly or mildly, explicitly or implic-itly. He can request or persuade, again strongly or mildly, explicitly orimplicitly. These appear as direct approaches. He can change B’s envi-ronment in a way that will bring about the desired action by B. Thisappears to be indirect. But in fact, all are indirect. A has no direct accessto any of B’s director systems. A can only get B to do something bymaking a change in B’s environment which sets in action one or more ofB’s director systems.

(Skemp, 1979, pp. 107–8)

Skemp goes on to point out that when motivation is seen in terms of what Adoes to B, it misses the complexity of B as a human being with will anddesires, and propensities to conform or to rebel. He claims that ‘a personwho is unaware that he has a choice, effectively has not (ibid., p. 110).

This applies as well to teachers and researchers as to learners.

Surprise and disturbance as motivation

Surprise: Nitsa Movshovits-Hadar

Nitsa Movshovits-Hadar (b. Israel, 1941–) is an educator and researcher.Here she expresses clearly and cogently a sentiment, which many teachershave experienced: the facts and relationships that make up mathematics arefull of potential surprise and delight and thus a positive form of disturbance(see p. 55, p. 101 and p. 161) motivating learning.

Intellectual surprise usually gives us a sense of fulfillment, an appreciationof some wisdom, a joy from its wittiness, and a drive to find some more.Making mathematical findings appear unexpected, or even as contra-

Affect in learning mathematics 101

expected, is the secret of teaching mathematics the surprise-way. Such away of teaching is not an end in itself, of course. It is however quite apromising means for achieving students’ interest, which has for long beenknown to be positively correlated with successful learning.

[ … ]Every single theorem can be turned into a surprise by considering the

unexpected matter which that theorem claims to be true. Sometimes thetheorem is so well known that it is hard to see the point. For examplewhat is so exciting about the claim that the sum of the interior angles of atriangle is 180 degrees?

To reach the surprise potential of a theorem it is usually helpful toassume we do not know it. Suppose we do not know that the sum ofthe interior angles of a triangle is 180 degrees. Would it be reasonableto suspect that all triangles, of any shape and size – equilateral, isos-celes, scalene, acute-angled, right-angled, obtuse-angled, very large(in area) and very small, narrow and wide – must all have the samesum for their interior angles? Would it not take a novice by surprise todiscover that the sum of the interior angles of triangles of variouskinds is a constant?

(Movshovits-Hadar, 1988, pp. 34–5)

Other examples she offers include Pythagoras’ theorem: if you did not knowit, what a challenge to find such a triangle; could there be a right-angledtriangle that did not satisfy the theorem? What relationships between squaresof sides characterises obtuse-angled and acute-angled triangles? Whathappens if you replace ‘squares’ on the sides by another shape, using thearea of similar shapes on the sides of the triangle? Notice the implicit use ofdimensions-of-possible-variation applied to Pythagoras’ theorem.

It is the mathematics teacher’s responsibility to recover the surpriseembedded in each theorem and to convey it to the students. The methodis simple: just imagine you do not know this fact. This is where you meetyour students. Let them examine their expectations, and make themrealize that they get new and very unusual results in every theorem.

(Movshovits-Hadar, 1988, p. 39)

Natural, conflicting and alien: Janet Duffin and Adrian Simpson

In an extension of Piaget’s notions of assimilation and accommodation (seep. 149), Festinger’s notion of cognitive dissonance (see p. 69), and of the roleof disturbance and surprise, two university teacher-researchers, Janet Duffin,who also evaluated the Calculator Aware Number Project (Duffin, 1996) andAdrian Simpson (b. Lincolnshire 1967–) suggested that examples andconcepts provided by teachers can be classified into three groups, accordingto the responses of learners:


Our theory postulates that learners encounter three kinds of experience,which we have come to call natural, conflicting, and alien, to which theyrespond by modifying their internal mental structures in different ways.We define a natural experience as one that fits the learner’s current mentalstructures and as one to which they respond by strengthening the struc-ture. A conflict is an experience which jars with the existing structures,either by showing that the way of working associated with the structurecannot cope in the way expected or by showing that two or more – previ-ously dissociated – mental structures can be brought to bear on the samesituation. We conjecture that learners respond to such conflicts by weak-ening or destroying the structure, by limiting the domain of experienceswith which the structure is expected to cope, or by constructing linksbetween previously separate structures. In contrast, an alien experience isone that has no fit with the existing mental structures at all: it neither fitsnor causes conflict. In response, a learner may ignore or avoid the experi-ence or may absorb it as a new, separate structure that then becomes liableto modification through later natural, conflicting, and alien experiences.

(Duffin and Simpson, 1999, p. 416)

There are similarities with assimilating and accommodating (see p. 149) andequilibration (see p. 148). See also p. 301 where Duffin and Simpson go onto explore what it means to understand in mathematics. Note resonanceswith Festinger’s cognitive dissonance (p. 69) and the role that such disso-nance plays in motivating learners to learn, and with the notion of fit at theheart of radical constructivism (see p. 93).

Ruptures and surprise: Alain Bouvier

Alain Bouvier (b. France, 1943–) is a mathematician and educator whoproduced a French dictionary of mathematics. In this extract he identifiesdisturbances or ruptures as crucial for development of mathematical ideas,both historically, and for learners.

It is possible to find [ruptures in mathematical thought] although itsoccurrences are not as well known as in the other disciplines.

• The most frequently mentioned rupture in mathematics concerns thediscovery of irrational numbers and the proof, by Pythagoras, of theirrationality of 2. Mathematics suddenly shifted from the idea that‘every number is a ratio of two whole numbers’ (as we may express itin modern terms) to the conception of two categories of numbers:rational and irrational.

[ … ]• Another famous rupture concerns the foundations of geometry.

After trying to prove Euclid’s fifth postulate, notably by developing


the consequences of various forms of its negation in the hope thata contradiction would appear, and after hiding these results fromthe scientific community, it eventually became necessary for math-ematicians to cross the Rubicon and allow new iconoclastic geom-etries to stand beside Euclidean geometry. The old frameworksuddenly exploded and within the framework of a plurality ofgeometries new problems arose, particularly the problem ofclassification.

• Without it ever having been made explicit, at the beginning of thenineteenth century it was understood that a continuous function waseverywhere differentiable, save possibly at a few exceptional points.To imagine, as Weierstrass did, that functions could be found whichare everywhere continuous but nowhere differentiable, was an enor-mous jump! Hermite refused to take any interest in these mathemat-ical objects, which he termed monsters. …

(Bouvier, 1987, pp. 18–19)

Surprise, disturbance, ruptures, are all manifested by the phrase ‘I have aproblem … ’, for the recognition of having a problem is evidence of aware-ness of something having altered without the possibility of a smooth equili-bration (to use the biological metaphor used by Piaget: see p. 148). Butbeing aware of ‘having a problem’ is a far cry from a list of questions or exer-cises or ‘problems’ in a textbook.

Hidden curriculum

Doing the tasks set by the teacher does not necessarily lead to success. Moreis expected, and more is required.

A template can be used mechanically to the extent of not learning from theexperience, of not making a meta-cognitive shift to make sense of theprocess and to see it as a process that can be repeated in other contexts. Inthe context of worked examples, the more comprehensive the workedexamples provided for the learners, the easier it is for learners to find amatching template, but the more likely they are to miss the intention, that is,of internalising the templates. In the context of example construction, if allthe requisite examples have been provided, learners have only a task ofsearching for the appropriate one.

Hidden curriculum: Benson Snyder

In a famous work which introduced the notion of a hidden curriculum (seep. 104). Benson Snyder, an American sociologist of education, observed thatlearners ‘played the game, read the cues, adapted to their immediate educa-tional circumstances’ (Snyder, 1971, p. xii).


[Students almost always translate what the faculty says they expect fromstudents] into a series of discrete, more or less manageable tasks whichthey infer is the actual basis for the grade their professors will give them.

… tasks then lead students to a set of tactics or maneuvers. Onestudent budgeted his time and commitment in a math course by doingthe last problem in the nightly homework set of six. He assumed itcontained all the necessary principles. Only if he had difficulty wouldhe then do the fifth, fourth, etc.. He saw the course as consisting ofthese ‘hurdles’ drawn up by the professor. He said this ‘exercise’ wouldprepare him well enough for the ultimate race – the examination. …

In effect, [students] are constantly asking questions about the differ-ences between the formal and hidden curriculum: What are the actualhurdles one must jump? … , and how is this reflected in the formal rulesand explicit descriptions? The answer to these and many similar ques-tions form the syllabus of the hidden curriculum. For most students, it ismore important than the visible curriculum …

(Snyder, 1971, pp. 6–7)

… To finish all the tasks of the formal curriculum would require far moretime than is available. In a typical coping pattern the student finds hemust neglect, selectively, certain aspects of the formal curriculum. Hemust learn what he can avoid doing, knowing where the risks areminimal and the cost is modest. The message is unstated, but it is as clearto the student as an item of information in the college catalogue.

(ibid., p. 12)

We have found repeatedly … that when the student’s sense of his worthis based principally on those narrow ranges of criteria of performancethat are used by the institution, two things follow. First, the student’sadaptation appears to be less likely to change, even in the face of newand different environmental pressures. Second, the student also appearsto be less aware of the consequences of having adjusted than are hisclassmates. To fit the real diversity of students we may not need to teachthem differently, we may just need to grade them differently.

(ibid., p. 159)

There are interesting resonances with the didactic contract and the didactictension (see p. 79). There are also close similarities with the notion ofdidactic transposition (p. 83), with Dweck’s findings concerning self-esteem,and with a social perspective on learning (see Maturana, p. 70). Callingfeatures hidden suggests intention, and leads some authors to want to makeeverything explicit to learners. However there are substantial, not to sayinsurmountable difficulties (see inner and outer aspects, p. 241).


Intended curriculum: The BACOMET group

BACOMET is an international group of mathematics educators that carriesout research on basic and fundamental topics of mathematics education. Thecomposition of the BACOMET group changes between projects but itsmembership remains at around 20 from about ten different countries. In thefirst project, Perspectives in Mathematics Education (1980) members ofBACOMET introduced ideas based on those of Vygotsky and colleagues setin the context of European mathematics education. Bent Christiansen(b. Denmark, 1921–1996) was an internationally respected researcher andorganiser who was influenced by Georges Papy (see p. 108) among others.Here he and his teacher colleague, G. Walther, start by talking about motiva-tion, and the difference between the intended and the actualised curriculum.

Students are motivated and initiated into activities by deliberate stepstaken by the teacher. These educational steps are planned in theperspective of educational purposes and intentions which more or lessexplicitly concern the learner’s acquisition of specified knowledge andknow-how shared with others, in short social knowledge and know-how. This intended learning is not limited to acquisition of closed end-products but should comprise process and product as complementaryaspects. It should contain learning of different types and at differentcognitive levels, and it should benefit from high priority on actionsdirected by the specific goal to learn.

… the problem is to identify means by which the teacher maypromote a unified conception – within the learner – of the role of task-and-activity, of learning, of mathematics, and of his personal, consciouscontrol of his own learning process.

(Christiansen and Walther, 1986, p. 264)

There are similarities with the teaching dilemma (see p. 96).

… learning cannot take place through activity performed by an individualin isolation, but must unfold in relation to activity mediated by otherpersons – the teacher, the parents, the peer group, etc. – and often throughactivity performed by a group including the individual in question.

(ibid., p. 267)

We list … five questions, offer brief comments, and ask the reader totake these questions a starting points for his own reflections.

The context of the task? Is the task concerned with internal mathemat-ical relationships? Or is it an application of a pro-forma type, such as thetraditional ‘word-problems’? Will the task fit properly into the teaching/learning process which is in progress in the class? And will it be ofappropriate interest and relevance for the students?


The complexity of the task? Will the solution of the task be established bymeans of a few rather obvious steps? Or will the performance call forseveral series of actions? And must these series be performed by the studentin some definite order (to be identified, perhaps, by himself in the course ofthe activity on the task)? Is the demand for a logical analysis of the task?

The degree of openness? Is the task described in an open form as in Howmuch does it cost to keep a dog?, or is a high degree of guidance inherent inthe textual formulation … ? Are the objects to be worked upon givenclearly and explicitly? Are some of the possibilities to be investigatedmentioned in the text? Are opening examples provided? Is an appropriatecontext set for further independent decisions and activity?

The form and appearance of the task? This question is about the formand effect of the technical presentation of the task. Thus our interest liesat this stage on an analysis of the ways in which the ‘task as text’ presentsitself to the student. …

Another important aspect is to what extent the text contains incite-ment to and starting points for reflection – or even for a dialoguebetween the reader and the text or ‘within’ the reader. And a third aspectfor analysis is, whether the text is challenging to an appropriate degree.

(ibid., pp. 276–7)

Christiansen and Walther go on to consider the task as a component of alarger ‘system’:

The structure of human activity is determined by a complex system ofmutually related factors: (1) the object for the activity and the inherentconditions; (2) the motive of the activity and the goals of the actions bywhich it proceeds; (3) the internal conditions and resources of the actingsubject; and (4) the external frames of the activity. …

As regards (1), the object (the task) is only to some extent directlyavailable for the student. It must be created for him or mediated to himand this brings about an initial state of affairs which differs from that of‘natural’ human activity.

As regards (2), needs, motive, and object are connected closely ingenuine human activity. Whereas, in the case of an educational task,questions arise about the extent to which the motive for the envisagedactivity is inherent in the task.

Finally, in relation to (3) and (4), the task as presented serves to initiatedifferent students in varying degrees and extents to the development ofactions which are potentially inherent in the envisaged activity.

[ … ]When task-and-activity is taken as the basic vehicle for learning, the

following three factors assume high importance: (i) the student’s concep-tion of tasks in school as tools for his own learning; (ii) the student’s


performance of actions ‘inherent in the task’ as needed for intendedlearning; (iii) the student’s personal control and evaluation of learning.

(ibid., pp. 290–1)

… three stages of interaction between teacher and learners in relation toa task selected or constructed by the teacher: (1) a stage of presentation;(2) a stage of independent activity individually or in groups; and (3) astage of concluding reflection. The activity of the students on the taskproceeds (in different forms) through these three stages. And the teacherperforms in each of these many different functions, although certainroles and functions have priority in each of the stages … .

(ibid., p. 293)

Authentic activity

Many authors have decried the artificiality of classroom activities. There aresharp contrasts between what children do in one context (for example,selling things on the streets in Brazil) and in school (see Nunes, p. 17; Lave,p. 75; Spencer, p. 115). This led to the notion of authentic activity as a moti-vational contribution: asking learners to engage in activities that mirror activ-ities carried out by adults in non-school contexts.

Personalising the curriculum: Jerome Bruner

Let me turn now to … the personalization of knowledge, getting to thechild’s feeling, fantasies, and values with one’s lessons. A generationago, the progressive movement urged that knowledge be related to achild’s own experience and brought out of the realm of empty abstrac-tions. A good idea was translated into banalities about the home, thenthe friendly postman and trashman, then the community, and so on. It isa poor way to compete with the child’s own dramas and mysteries.

[ … ]… to personalize knowledge one does not simply link it to the

familiar. Rather one makes the familiar an instance of a more generalcase and thereby produces awareness of it.

(Bruner, 1966, pp. 160–1)

There are similarities with Dewey’s notion of psychologising the subjectmatter (p. 45 and p. 203)

Authentic activity: Georges Papy

Georges Papy (b. Belgium, 1920–), was a teacher, educator and founder ofthe Belgian Centre of Mathematical Pedagogy. He promoted the use of the


language of sets for unifying mathematics, at the very youngest ages inschool. Much maligned in the 1960s because curriculum innovators wereunable to communicate the force and insight of Papy and others, his aimswere much the same as those advocating authentic tasks, though completelydifferently expressed and manifested.

From the foreword, by Howard Fehr, of Papy’s book, Modern Mathematics:

All of the concepts begin in everyday situations familiar to all students. If weare to teach more mathematics of the modern variety, to more students, andwe are to promote the desire of our education system to serve as a stairwayto fulfilling the needs and dreams of men, then surely it will be necessary tocultivate in some manner a genuinely modern global instruction in mathe-matics. This book serves as a great step forward in this direction.

Today, in our schools we are under great pressure to teach moremathematics to students at an earlier age because of the steady growth inthe applications of mathematics. At the same time, we are more sensitiveto the psychology of learning as related to mathematics, about which,however, our knowledge is limited. These factors make the problem ofcreating a unified modern syllabus a very complex one. On the otherhand today we have a greater understanding and deeper insight into theconcepts and theories of contemporary mathematics. It is this knowl-edge which places us in a good position to produce a unified syllabus.As one looks at the structures of mathematics, it is clearly seen that thesubject has acquired a unity, largely through the use of set theory. Nolonger do we think of mathematics as a collection of disjoint branches –arithmetic, algebra, geometry, analysis – having no inner relations toeach other. We think of it as a set of structures, all intimately related, andall applicable to many diverse situations.

(Papy, 1963, pp. v–vii)

And from Papy himself:

Mathematics, which at the beginning of the century had few applicationsoutside physics and engineering, has become a fundamental element ofcontemporary life and an indispensable tool in most spheres of thought,including science and technology. It has therefore become necessary toteach the basic elements of modern mathematics to all secondary schoolpupils, since any one of them may find himself in need of it.

Happily the development of mathematics and the variety of its appli-cations have profoundly modified the science and given it a morehumane appearance.

The advances made during the last hundred years have completelytransformed mathematics so that it now appears more familiar, moreintelligible, more precise, more accessible and more interesting.

Previously, elementary mathematics teaching could only deal with


artificial situations in which pieces of technical work were mixed in withvague (and usually not explicit) appeals to intuition.

Today a quite different approach is possible, in which the student isencouraged to take an active part in the building of the mathematicaledifice starting from simple, familiar situations.

Such is the object of this book, which is aimed at all those who wish tobe initiated into the mathematics of today, whatever their age and what-ever their previous development.

(ibid., preface p. ix)

Authenticity and the zone of proximal relevance

While some teachers focus on setting tasks in authentic contexts (Brown et al.,1989), some in contexts that enable direct action (Frankenstein, 1989; Mellin-Olsen, 1987), some in contexts that fit with learners’ immediate experience, andothers are content to work on abstracted mathematical objects removed fromcontexts until the techniques are mastered, a middle ground is looking forsettings and contexts that can ‘become real’ for learners. The Dutch RealisticMathematics project (see Gravemeijer, 1994) is not centrally concerned withwhat learners are specifically interested in, but rather with what they mightbecome interested in if their attention is suitably directed. Thus every learner hasthe power to imagine, and through exercise of their mental imagery canbecome interested and intrigued in questions that are not immediately practical.The term zone of proximal relevance is useful in order to refer to settings andtasks that, although not already of immediate relevance to learners as they seethings, can become ‘real’ for them through the use of their power to imagine.

Teacher desire

Concern about learner motivation leads teachers to exercise their owndesires that learners actually learn something. Learners can be motivated byteachers who display a love of and enthusiasm for their subject. Learners canalso be turned off by a teacher who cannot resist probing, and explaining,not letting learners internalise ideas for themselves.

Teacher lusts: Mary Boole

Finding it hard to wait for learners to think about and respond to questions isone version of what Mary Boole called teacher lusts, a strong term, but oftenappropriate. Desire to explain, and desire that learners enjoy their mathe-matics and appreciate what the teacher has appreciated serve as barriersrather than supports for teaching.

The teacher (whether school-teacher, minister of religion, politicalleader, or head of a family) has a desire to make those under him


conform themselves to his ideals. Nations could not be built up, nor chil-dren preserved from ruin, if some such desire did not exist and exertitself in some degree. But it has its gamut of lusts, very similar to thoserun down by the other faculties. First, the teacher wants to regulate theactions, conduct, and thoughts of other people in a way that does noobvious harm but is quite in excess both of normal rights and of practicalnecessity. Next, he wants to proselytise, convince, control, to arrest thespontaneous action of other minds, to an extent which ultimately defeatsits own ends by making the pupils too feeble and automatic to carry onhis teaching into the future with any vigour. Lastly, he acquires a sheerautomatic lust for telling other people ‘to don’t’, for arresting sponta-neous action in others in a way that destroys their power even to learn atthe time what he is trying to teach them. What is wanted is that weshould pull these three series tight so as to see their parallelism, and notgo on fogging ourselves with any such foolish notion as that sex-passionis the lust of the flesh and teacher-lust a thing in itself pure and good,which may legitimately be indulged in to the uttermost.

(Tahta, 1972, p. 11)

Explaining: John Holt

Explanations: We teachers – perhaps all human beings – are in the gripof an astonishing delusion. We think that we can take a picture, a struc-ture, a working model of something, constructed in our minds out oflong experience and familiarity, and by turning that model into a stringof words, transplant it whole into the mind of someone else. Perhapsonce in a thousand times, when the explanation is extraordinarily good,and the listener extraordinarily experienced and skillful at turning wordstrings into non-verbal reality, and when explainer and listener share incommon many of the experiences being talked about, the process maywork; and some real meaning may be communicated. Most of the timeexplaining does not increase understanding, and may even lessen it.

(Holt, 1967, p. 178)

Alertness and passivity: Mary Boole

Suppose I’m teaching, say, the process of multiplication. There are twothings which the pupils can get out of my instruction: (A) skill inperforming the operation of multiplication itself; and (B) a little of thepower to find out for themselves how to do other arithmetical opera-tions. Every process that I teach ought to be so taught as to add some-thing to the pupil’s chance of someday making out a rule for himselfwithout the aid of a teacher.

If we add together all the As of a child’s arithmetical career, theyconstitute … the body of his arithmetical knowledge; if we sum up the


Bs, they constitute what is called its life. The sum of the combined Aelements constitutes the ability to reckon the bulk or number of deadmaterial and to keep accounts according to any system chosen by anemployer. The sum of the B elements gives the extra power of bringingone’s knowledge to bear in forming a sound judgment on problemsconnected with living forces … .

Now the A element in any mathematical lesson can be imparted whilethe class is alert and eager; the B element cannot be imparted exceptunder the peculiar condition called by some mystic writers ‘Silence inthe soul’ awaiting further Light.

The two states, the alert and the passive, alternate in any good educa-tional regime; the alert phases being very much the longest, thepassively recipient ones short but quite undisturbed.

But under stress of competition the passive mystic phases of study arebeing crowded out. The reason is that England is so saturated with thespirit of advertisement that, in any given committee, the majority arealmost sure to be against the teaching of anything for which there isnothing to show at the next forthcoming examination.

(Tahta, 1972, p. 22)

Learner self-esteem

Not all activity produces learning. In the context of the didactic contract (see p.79), Guy Brousseau and Michael Otte identify some learner responses to chal-lenges intended to motivate learners to understand more deeply. They observethat some learners take on challenges but become immersed in details of thechallenge; others prefer to avoid challenge altogether wherever possible.

Altogether, one may in fact observe the inclination of certain pupils totake on willingly the first moment, namely the questions and problems,the uncertainty and openness, the complexity and the playing on theverge of knowledge. But they cannot quit or leave this gambling,autotelic, unrewarded behaviour and cannot get involved in a reallyresponsible manner … .

For others, it is impossible to accept the first moment. Problems andopen questions are not tolerable for them. … A question without animmediately conceivable answer causes anguish for them. Theyobsessionally ask the teacher for answers, for decision procedures toreach an answer, for algorithms.

(Brousseau and Otte, 1991, pp. 33–4)

Learner theories: Carol Dweck

Carol Dweck (b. New York, 1946–) is an American psychologist who hasfocused on the issue of self-esteem in learners. In Dweck (1999) she


summarises for teachers a lifetime of work with numerous colleagues,making specific suggestions for working with learners with low self-esteem.Here she distinguishes two theories learners have about themselves, andtheir implications for learning.

Some people believe their intelligence is a fixed trait. They have acertain amount of it and that’s that. …

This view has repercussions for students. It can make students worryabout how much of this fixed intelligence they have. … They must looksmart and, at all costs, not look dumb.

[To feel smart they need] easy, low-effort successes, and outper-forming other students. Effort, difficulty, setbacks, or higher-performingpeers call their intelligence into question … .

Challenges are a threat to self esteem. …[Lavish well-meaning praise for very little fosters] an over concern for

looking smart, a distaste for challenge, and a decreased ability to copewith set-backs.

[Other people believe intelligence is] … something they can cultivatethrough learning. … [Everyone], with effort and guidance, can increasetheir intellectual abilities. …

It makes them want to learn … Why waste time worrying about whetheryou look smart or dumb, when you could be becoming smarter? …

[ … ]Self-esteem … is a positive way of experiencing yourself when you

are fully engaged and are using your abilities to the utmost in pursuit ofsomething you value.

(Dweck, 1999, pp. 2–4)

For learners with low self-esteem convinced that their intelligence is limitedand fixed (‘I can’t so I won’t’), it is natural to try to persuade the teacher to givethem less challenging ‘easier’ work ‘to ease them along’. This minimises thepossibility of being brought to the edge of understanding, and so to revealinghow little they feel they know. Learners identified as ‘low-attaining’ and puttogether in a group are well aware that they are being given ‘simpler tasks’.

The teacher, meanwhile, is eager to develop learners’ self-esteem and sosimplifies the tasks in order to find a level at which the learners can operatesuccessfully. The aim is to convert them to an inner language of ‘I can’, withtheir intelligence something that grows in response to challenge, and sogradually to restore the challenge level of the work. Unfortunately, this ideal-ised process usually stalls, as it is in the learners’ short-term interests to keepthe work as simple as possible without it becoming overtly ‘baby-ish’.

Note the parallels with funnelling (see p. 274): there the questions aremade simpler until the learner feels safe in answering. Here the tasks aremade simpler until the learners finally engage. Dweck suggests converting ‘Ican’t so I won’t’ into ‘I can and I will’ by replacing ‘I’m at my limits’ by ‘I can


try harder’. This proves to be a useful contribution to the perhaps inevitablesimplifying of tasks, and a powerful device for re-invigorating depressedlearners. The teacher’s art is in simplifying or altering just enough so thatlearners can feel success and can experience improving their thinking, so asto reactivate their motivational desires and interest hand in hand withconversion to ‘I can and I will’. Notice also the similarity with the didacticcontract (see p. 79).

Learners can all too easily develop a habit of mind that they do not under-stand, even when they palpably respond appropriately. Some people seeksupport by justifying tasks through authentic contexts and authentic tasks, orthrough real problem-solving. But by the time learners have reached thepoint where they are convinced that they will not succeed, that they ‘can’t domaths’, contexts are merely a diversion.


Experiences can be natural, conflicting or alien. There is widespread agree-ment that appropriate disturbance, disruption of expectation, is what initiatesand motivates acting in such a way as to make sense, and to learn.

Learners need encouragement and personal control over the amount ofchoice and freedom being dealt with at any one time. Some learners are moti-vated more by effectiveness and success in the material world (authenticity ofactivity), some are motivated more by aesthetics of patterns and structures.

Teachers’ excessive desire that their learners learn and develop may in somecases be part of the barrier to that learning; teachers’ low expectation may simi-larly blunt learners’ motivation.

5 Learners’ powersLearners’ powers

Introduction

In this chapter, the extracts relate to the notion that learners make use oftheir natural powers to make sense of the world in general, and that these area resource to be developed by the teacher of mathematics. The chapter startswith a collection of extracts about these natural powers in general and subse-quent extracts look at particular examples.

Natural powers

In his comprehensive advice to teachers, J. Calkin (1910) summed upfrom a Canadian perspective a sentiment which seems to have beenpervasive in educational circles in many countries, even if not in practicein schools. ‘ … the mind is a power to be developed rather than a recep-tacle to be filled is a sound maxim in education’ (Calkin, 1910, p. 18).

Herbert Spencer observed what parents and teachers have alwaysobserved:

… ‘that children in the household, the streets and the fields’ (Spencer,1911, p. 24) learn all kinds of things effortlessly, with eager pleasure, yetthese same children often have great difficulty learning quite elementarythings in formal educational settings.

(Egan, webref a)

There are interesting connections with the work of Nunes and colleagues(1993) on the facility of young children selling things in the streets in contrastto their performance in a school setting. This theme was taken up by Lave(1988) and then by Seeley Brown and colleagues (1989) who developed thenotions of authentic mathematics (see p. 108) and learning through appren-ticeship (see p. 75).

Powers: Herbert Spencer

Spencer proposed that children are naturally inquiring, constructing, andactive beings, so the developing powers of children provide the basis for hiseducational philosophy (see also p. 35).

Who, indeed, can watch the ceaseless observation and inquiry and infer-ence going on in a child’s mind, or listen to its acute remarks on matterswithin the range of its faculties, without perceiving that these powers itmanifests, if brought to bear systematically upon studies within the samerange, would readily master them without help? This need for perpetualtelling results from our stupidity, not the child’s. We drag it away from thefacts in which it is interested, and which it is actively assimilating of itself. …

(Spencer, 1878, p. 72)

Kieran Egan (b. Ireland, 1942–) is now a Canadian researcher with anunusual perspective on education. He accuses Spencer of a negative influ-ence on modern education:

In this paper I will argue that the conception of education that continuesto shape our schools, and influences what we do to children in itsname, was given its modern sense as a result of ideas that were largelyformulated in the 1850s. I will try to show the source of many of ourpresent most generally held beliefs about learning, development, andthe curriculum, and show that they were based on ideas that were,simply, wrong. These ideas continue to be the source of catastrophicdamage and waste of life, and are responsible for the general ineffective-ness of schooling.

In describing a catastrophe one needs an appropriate villain, and thebest villain of modern education is Herbert Spencer. … his educationalideas, based on general principles shown to be false, became the rarely-questioned basis of modern education.

(Egan, webref a)

Egan points out that Spencer coined the expression ‘evolution’, which wassnappier than Darwin’s original ‘descent with modifications’ or ‘naturalselection’. Spencer also coined the term ‘survival of the fittest’ which Darwinlater took up in a limited fashion, while Spencer applied it to all andeverything.

Powers: Bertrand Russell

Bertrand Russell (b. Wales, 1872–1970) was a mathematician and philosopher.As mathematician he worked with Whitehead to try to place mathematics on afirm foundation of logic; as a philosopher he promoted sensitivity to feelings


as well as rational and deductive thinking. Here he summarises a long descrip-tion of how early education was based on the widespread practice of trying toforce learners to learn through corporal punishment.

The spontaneous wish to learn, which every normal child possesses, asshown in efforts to walk and talk, should be the driving-force in educa-tion. The substitution of this driving-force for the rod is one of the greatadvances of our time.

(Russell, 1926, p. 25)

Discovery: Alfred North Whitehead

Here Whitehead encompasses issues which are still with us, including utility(see authenticity, p. 108).

Let the main ideas which are introduced into a child’s education be few andimportant, and let them be thrown into every combination possible. Thechild should make them his own, and should understand their applicationhere and now in the circumstances of his actual life. From the very begin-ning of his education, the child should experience the joy of discovery. Thediscovery he has to make, is that general ideas give an understanding ofthat stream of events which pours through his life, which is his life.

(Whitehead, 1932, p. 3)

Whatever interest attaches to your subject matter must be evoked hereand now; whatever powers you are strengthening in the pupil, must beexercised here and now; whatever possibiltities of mental life yourteaching should impart, must be exhibited here and now. That is thegolden rule of education, and a very difficult rule to follow.

(ibid., p. 9)

Thinking: Max Wertheimer

Max Wertheimer (b. Prague, 1880–1943) was one of the major thinkers andforces behind the elucidation of gestalt psychology in Germany. He left forthe USA in 1933 where he was professor of psychology and philosophy inthe New School for Social Research in New York. Writing in the 1930s,Wertheimer is describing the components of thinking, first from what wewould call an experiential or phenomenological point of view, and then,extracted here, from a behavioural point of view.

Thinking consists in

• envisaging, realizing structural features and structural requirements;proceeding in accordance with, and determined by, these requirements;

Learners’ powers 117

and thereby changing the situation in the direction of structural improve-ments, which involves:

• that gaps, trouble-regions, disturbances, superficialities, etc., be viewedand dealt with structurally;

• that inner structural relations – fitting or not fitting – be sought amongsuch disturbances and the given situation as a whole and among itsvarious parts;

• that there be operations of structural grouping and segregation, ofcentering, etc.;

• that operations be viewed and treated in their structural place, role,dynamic meaning, including realization of the changes which thisinvolves.

(Wertheimer, 1961, pp. 235–6)

Note that Wertheimer makes use of the notion of disturbance (see p. 55) asunderlying or producing learner action. Although he expresses himself interms of thinking actions, it is clear that he thinks people have the requisitepowers to act in these ways.

• realizing structural transposability, structural hierarchy, and sepa-rating structurally peripheral from fundamental features – a specialcase of grouping;

• looking for structural rather than piecemeal truth.

In human terms there is at bottom the desire, the craving to face the trueissue, the structural core, the radix of the situation; to go from anunclear, inadequate relation to a clear, transparent direct confrontation –straight from the heart of the thinker to the heart of his object, of hisproblem. All the items hold also for real attitudes and for action, just asthey do for thinking processes.

(ibid., p. 236)

Wertheimer then suggests that it is perfectly possible to rephrase theseobservations in terms descriptive of what behaviour an observer mightnotice, contrasting superficial and structural awareness, in terms of: compar-ison and discrimination (identification of similarities and differences); anal-ysis (looking at parts); induction (generalisation, both empricial andstructural); experience (gathering facts or vividly grasping structure); experi-mentation (seeking to decide between possible hypotheses); expressing‘one variable is a function of another variable’; associating (items togetherand recognising structural relationships); repeating; trial and error; andlearning on the basis of success (with or without appreciating structuralsignificance) (based on Wertheimer, 1961, pp. 248–51).


Powers: Mary Boole

Mary Boole (b. England, 1832–1916) was brought up in France beforereturning to England. She edited the books and papers of her husband(George Boole, creator of Boolean Algebra) until his death. Mary then taughtmathematics and wrote extensively about teaching, as well as pursuing herinterests in Eastern religious thought.

Some extracts from her extensive educational writings can be found in acollection made by Dick Tahta (1972, see also p. 110).

My husband told me that when he was a lad of seventeen a thought struckhim suddenly, which became the foundation all his future discoveries. Itwas a flash of psychological insight into the conditions under which a mindmost readily accumulates knowledge. Many young people have similarflashes of revelation as to the nature of their mental powers; those to whomthey occur often become distinguished in some branch of learning; but tono one individual does the revelation comes with sufficient clearness toenable him to explain to others the true secret of his success.

(Boole, 1901, p. 951 (from a letter written in 1901))

Powers: Vadim Krutetskii

Krutetskii was a Soviet psychologist, deputy director of the research Instituteof General and Educational Psychology at the USSR Academy of PedagogicalSciences, and head of the section on abilities of the Vygotsky school. Hismonumental study of mathematical ability has influenced generations ofresearchers. Here he traces some of the background to his own work andthat of his colleagues (see activity theory, p. 84).

Soviet psychology resolves, from a Marxist position, one of the mostcomplicated issues in the psychology of ability: the relationship betweenthe innate and the acquired in ability. A basic tenet of Soviet psychologyon this issue is the thesis that social factors have a decisive value in thedevelopment of abilities; that the leading role is played by man’s socialexperience – by the conditions of his life and activity. Mental traitscannot be inborn. This can be said of abilities as well. Abilities arealways the result of development. They are formed and developed inlife, during activity, instruction, and training.

… Man is endowed at birth with only one ability: the ability to formspecifically human abilities. …

[ … ]Abilities cannot be inborn; only inclinations for abilities – certain

anatomical and physiological features of the brain and nervous system –are present at birth.

(Krutetskii, 1976, pp. 60–1)


Krutetskii isolates seven assumptions underlying his research; here are fourof them:

• Abilities are always abilities for a definite kind of activity; they existonly in a person’s specific activity. Therefore they can show up onlyon the basis of an analysis of a specific activity. Accordingly, a mathe-matical ability exists only in a mathematical activity and should bemanifested in it.

• Ability is a dynamic concept. It not only shows up and exists in anactivity but is created and even developed in it. Accordingly, mathe-matical abilities exist only in a dynamic state, in development; theyare formed and developed in mathematical activity.

• At certain periods in a person’s development, the most favorableconditions arise for forming and developing individual types ofability, and some of these abilities are provisional or transitory. …

• Progress in a mathematical activity depends not on an ability takenseparately, but on a complex of abilities.

(ibid., pp. 66–7)

Powers: Maria Montessori

Maria Montessori (b. Italy, 1870–1952) was the first female medical graduatefrom the University of Rome. Frustrated by failings of the education systemand deeply concerned about the poverty she saw around her, she wasconvinced that children could learn to read and write if only they were put inan environment in which it was useful to them to learn. Her methodsinvolving learners taking responsibility for choosing, using, and puttingaway structured apparatus has influenced primary education everywhere.She advocated observing children in order to learn how to improve teaching:she herself called ‘the discovery of the child’ her greatest contribution.

According to Maria Montessori, ‘A child’s work is to create the personshe will become’. To carry out this self-construction, children haveinnate mental powers, but they must be free to use these powers. Forthis reason, a Montessori classroom provides freedom while maintainingan environment that encourages a sense of order and self-discipline.‘Freedom in a structured environment’ is the Montessori dictum thatnames this arrangement.

Like all thinkers in the Aristotelian tradition, Montessori recognizedthat the senses must be educated first in the development of the intellect.Consequently, she created a vast array of special learning materials fromwhich concepts could be abstracted and through which they could beconcretized. In recognition of the independent nature of the developingintellect, these materials are self-correcting – that is, from their use, thechild discovers for himself whether he has the right answer. This feature


of her materials encourages the child to be concerned with facts andtruth, rather than with what adults say is right or wrong.

Also basic to Montessori’s philosophy is her belief in the ‘sensitiveperiods’ of a child’s development: periods when the child seeks certainstimuli with immense intensity, and, consequently, can most easilymaster a particular learning skill. The teacher’s role is to recognize thesensitive periods in individual children and put the children in touchwith the appropriate materials.

[ … ]… Dewey’s concern was with fostering the imagination and the devel-

opment of social relationships. He believed in developing the intellectlate in childhood, for fear that it might stifle other aspects of develop-ment. By contrast, Montessori believed that development of the intellectwas the only means by which the imagination and proper social relation-ships could arise. Her method focused on the early stimulation andsharpening of the senses, the development of independence in motortasks and the care of the self, and the child’s naturally high motivation tolearn about the world as a means of gaining mastery over himself and hisenvironment.

(Enright and Cox, webref)

The focus on self-realization through independent activity, the concernwith attitude, and the focus on the educator as the keeper of the environ-ment (and making use of their scientific powers of observation andreflection) – all have some echo in the work of informal educators.However, it is Maria Montessori’s notion of the Children’s House as astimulating environment in which participants can learn to take respon-sibility that has a particular resonance.

(Smith, M., 1997)

Powers: Caleb Gattegno

In the postscript to one of his less philosophical books, Gattegno summariseshis thinking about learners’ powers. The development of children’s powersand making use of them in teaching formed the backbone of all Gattegno’sinvestigations, so it is a theme which he returns to again and again.

What is it, then, that will allow us to teach mathematics to anyone with afunctioning mind and an inclination to learn? Simply, finding a way tomake the learner aware of the powers of his mind – the powers he usesevery day, those which allowed him to learn his native language and touse imagery and symbolism. This means that the job of teaching is oneof bringing about self-awareness in learners through whatever meansare available in the environment: words, actions, perceptions of transfor-mations, one’s fingers, one’s language, one’s memory, one’s games,


one’s symbolisms, one’s inner and outer wealth of perceived relation-ships, and so on.

(Gattegno, 1974, p. 111, postscript)

Young children are continually investigating their powers of perceptionand action; they use these powers spontaneously – without having to betaught – in order to elaborate their experience of themselves and of theworld. … this can become part and parcel of mathematical education.

(Gattegno, 1988, p. 165)

Teachers of children will say that the greatest power of the mind is thecapacity to transform. Anyone who speaks and speaks properly – asmany two year-olds can do very easily – can transform according to hisperception of the situation and according to the criteria that he hasmastered and understood.

(Gattegno, 1970, p. 23)

Powers: Dick Tahta, Bill Brookes and David Wheeler

Brookes, Tahta and Wheeler were early members of what became the Asso-ciation of Teachers of Mathematics which was initiated by Gattegno. Theymet regularly to observe lessons together and discuss what they saw.

If in our enthusiasm for providing active experience for young childrenwe do no more than provide the springs and balances, the sand andwater, with requests for recording of what happens in certain selectedsituations, then we run the danger of abdicating from mathematics alto-gether. We certainly encourage the same abdications if we think in termsof children ‘discovering’ relations in certain external situations. Some ofthe most important mathematics relations stem from the earliest mentaland emotional activity of the infant. We make sense of our environmentby imposing these relations upon it. In developing our understandingand control of these relations in this way we further provide the possi-bility of a control of the environment. In order to develop the fullestresources of the human mind, it may be more important to think ofcreating mathematics rather than discovering it. In the creation of likesand unlikes we detect ‘the mind at work creating works of the mind’.And this is mathematics.

(Tahta and Brookes, 1966, p. 8)

There are similarities in their concern with a dilemma raised by Ainley (seep. 250).


Powers: Colin Banwell, Dick Tahta and Ken Saunders

Banwell, Tahta, and Saunders were teachers and teacher educators in Devonwhen they wrote an immensely practical book which inspired generations ofteachers to move towards teaching investigatively, engaging learners inthinking and exploring mathematically. This brief extract is taken from theirsummary.

In developing and using the powers that he already has there is no otherfeedback than his own judgement and descision, though these may beaffected by the conventions of social agreement.

(Banwell et al., 1972, p. 61)

They also wrote ‘Everyone is a mathematician though he may not know this.The mind knows more than it knows it knows.’ (Banwell et al., 1972(updated 1986), p. 61.) We suggest that no better summary of the power ofhuman minds can be given.

Powers: Richard Skemp

Skemp was similarly impressed with what very young children achieve.

An infant aged twelve months, having finished sucking his bottle,crawled across the floor of the living room to where two empty winebottles were standing and stood his own empty feeding bottle neatlyalong side them. A two-year-old, seeing a baby on the floor, reacted to itas he usually did to dogs, patting it on the head and stroking its back.(He had seen plenty of dogs, but had never before seen another babycrawling.)

In both these cases the behaviour of the children concerned implies:… some kind of classification of their previous experience; [and] thefitting of their present experience into one of these classes.

We all [behave like this] all the time; it is thus that we bring to bear ourpast experience on the present situation. The activity is so continuousand automatic that it requires some slightly unexpected outcomethereof, such as the above, to call it to our attention.

At a lower level, we classify every time we recognize an object as onewhich we have seen before. On no two occasions are the incomingsense data likely to be exactly the same, since we see objects at differentdistances and angles, and also in varying lights. From these varyinginputs we abstract certain invariant properties, and these propertiespersist in memory longer than the memory of any particular presentationof the object.

(Skemp, 1971, pp. 19–20)


Note the passing reference to disturbance (see p. 55) and particularly the useof imagery (see p. 129).

Powers: John Mason

John Mason (b. Ontario, 1944–) was profoundly influenced in his teachingby seeing a film in 1967 of George Polya teaching some undergraduates, byyear-long contact with the mathematician and teacher J. G. Bennett (1897–1974), and by contact with Gattegno. This book arose from his growingawareness that themes he had been promoting had actually beenpropounded in earlier generations. Here he advocates thinking in terms oflearners’ powers.

Every child that gets to school has already exhibited amazing powers.They have decoded the sounds that adults make, and learned to fashionsounds that others make sense of. They have learned to coordinate grossand fine muscles in order to move about, to pick things up and put themdown. They have worked out that unsupported things fall, that somethings are heavy and others light, with some even rising all by them-selves unless held down. They have worked out that there are patternsto light and dark, to adults’ presence and absence. They have learned torecognise people despite changes in hair shape and colour, clothes,glasses, etc.. Indeed, an amazing intelligence has been displayed. Theyhave nascent theories about people (and when people are happy, sad,irritated, angry, etc.).

Then they come to school. We try to formalise some of what they cando intuitively (speech becomes print, ordering and counting becomenumbers, talk becomes writing). In the process we do not alwaysmanage to draw upon those existing powers as fully as we might, withthe result that many children act as if they believe those powers are notactually wanted in the classroom.

(Mason, 2001a)

Every learner comes to school having already demonstrated tremendouspowers of sense-making. Specific powers are described in the followingsections.


Learners have natural powers with which they make sense of the world ingeneral. These are a valuable resource to be developed by the teacher ofmathematics.

Discerning similarities and differences

The power to distinguish, to discern, to make distinctions is of course centralto education. However, authors who draw attention to it imply that learnersare not always encouraged to use this power for themselves.

Discernment: Thyra Smith

Many authors have marvelled at the powers of young babies to make senseof the world in which they suddenly ‘discover themselves’. Smith became anHer Majesty’s Inspector of Schools (HMI) and received an OBE for herservices to education. Her short but immensely practical book begins withobservations of a baby. She goes on through the book to describe the child’svarious powers and how they develop through experience of the world.

A young baby can suck and grasp and move; but he does not appear atfirst to know what he grasps or sucks or moves. At a very early stagehowever he knows whether what he sucks is satisfying or otherwise; thisis probably the beginning of discrimination.

During the first months his waking periods are occupied in fairlycontinuous motion; but his movements appear random. They bring himin contact with things that are grasped or sucked or pushed away androlled about in a fashion that suggests that he is aware of their presencebut not of their nature.

(Smith, 1954, p. 1)

Perhaps it is useful to summarize very generally what arithmetical andmathematical conceptions are gained by the young child in a natural andpractical way.

He is aware of self, and not-self, that is of ‘I’ and of other things thathave a life of their own. He is aware of ‘some’ and ‘a lot’ and ‘none’. Thelast named is possibly mere absence or lack which is expressed either byhunting to find things or by making requests to be supplied. ‘Bigness’and ‘Littleness’ are known as are ‘High’ and ‘Low’. None of these termsmay be used spontaneously, though they are evidently understood.Parts of the body are known and the difference between unity andplurality is appreciated practically and intuitively. …

Ideas of differences in shape are shown in reactions to balls that willroll, bricks that can be pushed together or placed on one another, boxesthat open and shut and so forth.

[ … ]Power of discrimination is shown by preferences shown for particular

toys. Things are hustled together or pushed apart to make groups ofvarying sizes without any consideration for the ‘size’ aspect of the matter


except that obviously he differentiates between what is enough to besatisfying or too few to be interesting.

[ … ]Knowledge at this stage is ‘sensed’ intuitively and we are aware of its

existence through observation and interpretation of the child’s actions.Much repetition and continued use will establish greater sureness andwhen the time is ripe the child will show that he is aware – that he knowswhat he knows. But thus far his basis is firmly set on concrete things; …

(ibid., pp. 9–10)

Note the similarity with theorems-in-action (see p. 63). Note also the influ-ence of Piaget’s research in describing different ways in which childrenencounter and engage with the world (see p. 92), and similarities with struc-ture of attention (see p. 60), and the role of practice (see p. 174).

Discernment: Caleb Gattegno

Another attribute a child brings with him is the ability to notice differ-ences and assimilate similarities. What does this mean? Aristotle put theability to perform this operation at the foundation of basic logic andevery child owns it: he brings it with him. Every child knows that thebasis of living is to recognize differences and similarities.

Of two cups of the same make, we can hold one cup with the handlein front, the other so that the handle is at the back. One may be pink, theother white. We still say both are cups, not two distinct kinds of objects,a pink object without a handle, a white object with a handle – but cupsand that there are two of them.

We would say otherwise if we could not ignore differences and findthe attributes that bring them together, as well as see the attributes thatseparate them.

(Gattegno, 1970, pp. 25–6)

Distinction making is the first thing we need to do as organisms, even beforewe are born. Distinction making then develops into more sophisticated waysof making sense of the world. See van Hiele (p. 59 and p. 163) and Maturana(p. 70) for example.

Stressing and ignoring: Caleb Gattegno

The very act of discerning involves stressing some sense impressions, ormore generally, some features, and ignoring others. As Gattegno says ‘Toisolate the mental activity called mathematics is as easy as to merge it withany other, for it is a property of the mind to isolate and merge, to stress andignore’ (Gattegno, 1970a, p. 136). He recognised that this fundamental action


which leads to discernment, to distinction making, and hence to classifica-tion, is the very basis for abstraction and generalisation.

Stressing and ignoring: John Dewey

Of course, once alerted to a distinction, it is possible to find it in earlier writers.Here is John Dewey on stressing and ignoring, which he calls selective emphasis.

The favoring of cognitive objects and their characteristics at the expenseof traits that excite desire, command action and produce passion, is aspecial instance of the principle of selective emphasis which introducespartiality and partisanship into philosophy. Selective emphasis, withaccompanying omission and rejection, is the heart-beat of mental life. Toobject to the operation is to discard all thinking. But in ordinary mattersand in scientific inquiries, we always retain the sense that the materialchosen is selected for a purpose; [but] there is no idea of denying what isleft out, for what is omitted is merely that which is not relevant to theparticular problem and purpose in hand.

(Dewey, 1938, pp. 24–5)

Dewey also points to the related concept of invariance in the midst ofchange (see p. 129) as lying at the heart of scientific enquiry.

[A scientist] seizes upon whatever is so uniform as to make the changesof nature rhythmic, and hence predictable. But the contingencies ofnature make discovery of these uniformities with a view to predictionneeded and possible. Without the uniformities, science would be impos-sible. But if they alone existed, thought and knowledge would be impos-sible and meaningless. The incomplete and uncertain gives point andapplication to ascertainment of regular relations and orders. These rela-tions in themselves are hypothetical, and when isolated from applicationare subject-matter of mathematics (in a non-existential sense). Hence theultimate objects of science are guided processes of change.

(ibid., p. 160)

John Dewey spoke elsewhere in terms of how experience of a ‘thing’ alreadyinvolves selection of certain attributes to attend to, with others beingignored, and that this is how ‘things’ come into existence for learners as‘things in themselves’. (See reification, p. 167).

Ideas … are suggestions. Nothing in experience is absolutely simple,single, and isolated. Everything experienced comes to us along with someother object, quality, or event. Some object is focal and most distinct, but itshades off into other things. A child may be absorbed in watching a bird;for the bright center of his consciousness there is nothing but the bird


there. But of course it is somewhere – on the ground, in a tree. And theactual experience includes much more. The bird also is doing something– flying, pecking, feeding, singing, etc.. And the experience of the bird isitself complex, not a single sensation; there are numbers of related quali-ties included within it. This highly elementary illustration indicates why itis that the next time a child sees a bird, he will ‘think’ of something elsethat is not then present. That is to say, a portion of his present experiencewhich is like that of prior experience will call up or suggest some thing orquality connected with it which was present in the total previous experi-ence; that thing or quality in turn may suggest something connected withitself; it not only may do so, but it will do so unless some new object ofperception starts another train of suggestions going. In this primary sense,then, the having of ideas is not so much something we do, as it is some-thing that happens to us. Just as, when we open our eyes, we see what isthere; so, when suggestions occur to us, they come to us as functions ofour past experience and not of our present will and intention. So far asthoughts in this particular meaning are concerned, it is true to say “itthinks” (as we say “it rains”), rather than “I think”. Only when a persontries to get control of the conditions that determine the occurrence of asuggestion, and only when he accepts responsibility for using the sugges-tion to see what follows from it, is it significant to introduce the ‘I’ as theagent and source of thought.

(Dewey, 1933, pp. 41–2)

Dewey is referring to what Skemp called resonance as the mechanismwhereby stimuli of the senses produces complex experiences which include‘thinking of something that is not then present’. The observation that ‘itthinks’ is more accurate than ‘I think’ has far reaching implications in thedevelopment of useful habits, and working against obstructive habits.

Same and different: Dick Tahta and Bill Brookes

Asking learners to look out for what is the same and what differs betweentwo or more objects, situations or phenomena proves remarkably fruitful atinvoking learners’ powers to stress and ignore, which lies behind the mathe-matical theme of invariance in the midst of change (see p. 193).

It is in these choices [of uses of words such as ‘same’] that the essence ofmathematics resides.

If certain choices about the use of the word ‘same’ lie at the heart ofmathematical activity then mathematics starts in the cradle, for a sense ofone’s own identity from one moment to the next is one of the firstlessons to be learnt. Mummy is quickly felt to be the same person whenshe comes and goes; it takes much longer for Daddy to acquire the sameinvariance. The rattle is seen from different positions and felt in different


ways. Each perception is different but as it is explored helps to build upthe idea of something permanent, something that is always the same.Later on it becomes more and more difficult to recapture the differencesbetween our perceptions. We very rarely see a circle when we look at apenny, but that is the shape we say that we see when we are asked. Inan agreed sense, a circle remains the same wherever we perceive itfrom; in another agreed sense it does not. In one case a circle and anellipse are the same but in the other they are different.

[ … ]It is significant that the use of the word ‘same’ reveals an incredibly

complicated sequence of changes of view-points during the time that it isbeing used. To recognise that a child uses this word in this manner is torecognise some of the capabilities for abstraction which he is often denied.

(Tahta and Brookes, 1966, pp. 4–5)

One of the principal themes of mathematics is invariance in the midst ofchange (see p. 193). The device of asking learners ‘what is the same andwhat is different’ and then negotiating what is worth attending to, has beenextensively exploited and researched by the team of Alf Coles (b. London,1970–) and Laurinda Brown (b. Yorkshire, 1952–). They found it to beextremely effective in promoting active learners of mathematics (Brown andColes, 1999; 2000; Coles and Brown, 1999).

Distinguishing things on the basis of stressing some features as beingdifferent, while treating others as ‘being the same’ is the basis of classification(see p. 135), as Skemp has already suggested. Putting things in order, comparingand contrasting, can be seen in the play of very young children, in the organisa-tion of collecting as adolescents, and in the structure of mathematics. We succeedas organisms because we can classify, that is, we can discriminate and recognisesimilarities and differences, which we do by stressing some features and ignoringothers. This ability to stress-and-ignore at the same time lies behind the variouspowers mentioned here: it is the basis for specialising and generalising, for imag-ining and expressing, and for ordering and classifying.

Mental imagery and imagination

It is through the power of mental imagery that we are able to be simulta-neously present and yet ‘somewhere else’; that we are able to enter theworld of mathematical images, and through that, the world signified bysymbols. It is the means by which we identify ‘things’ as ‘things’ that Deweymentioned in the previous extract.


Stressing and ignoring produces discrimination, leading to classification andrecognition. Using ‘what is the same and what different about … ’ as a peda-gogical device makes use of these powers.

Imagery: Caleb Gattegno

Every one of us knows of the fantastic things that can happen in ourdreams and nightmares. Looking both at the dynamics of imagery and athow it affects the content of our dreams, we can learn a great deal aboutwhat children bring with them to their mathematics studies.

The type of transformation met in this context, when the teacher callsupon mental evocations to advance mathematical understanding, is onethat remains in contact with mental energy, keeps some continuitybetween the initial and the final forms of the images (which are dynamic,as in dreams), and produces effects that display the algebras applied tothem. When concentrating on imagery, one is more aware of content thanof transformation and stresses images per se all through the process.

By asking students to shut their eyes and to respond with mental imagesto verbal statements enunciated by the teacher, one makes them aware:

• that in their mind imagery is connected with the rest of their experi-ence, and

• that in itself is a power.

Indeed this type of relationship between teacher and students can beused to generate whole chapters of mathematics. The key here is thedynamic attribute of imagery, which can be seen as being equivalent tocertain mathematical properties.

(Gattegno, 1970, pp. 26–7)

An excellent source of specific and practical tasks for this purpose can befound in Leapfrogs (1982). ‘Leapfrogs’ is the name for a group which met regu-larly to develop resources for mathematics classrooms designed to maximisethe potential for their use in different situations and for different purposes.

Here Gattegno is referring to the use of mathematical animations and thenlinks algebra and imagery:

Because images are dependent on our will, once we begin deliberatelyto employ them, we can very soon obtain an awareness that indeedimagery is a power of the mind, and it can yield in a short time vastamounts of insights into fields that become almost sterile when thedynamics are removed from them … .

Algebra is present in all mathematics because it is an attribute of thefunctioning mind. Imagery is present at will and can remain presentwhile the mind is at work on it or on some elements within it.

Who can doubt that many more children will be at home with mathe-matics when features of it are presented to them as the recognition ofwhat one can contemplate within one’s mind when it is responding tomental stimuli.

(ibid., pp. 27–8)


Imagination: Kieran Egan

Egan views teaching as a form of story telling (see also Bruner, p. 68), andpoints to the importance of imagination, even if it is difficult to get hold of.

A continuing theme of this book (Egan, 1986) is that children’s imagina-tions are the most powerful and energetic learning tools. Our most influ-ential learning theories have been formed from research programs thathave very largely focused on a limited range of children’s logicalthinking skills. That research has largely neglected imagination, becauseimagination is, after all, difficult stuff to get any clear hold on. Conse-quently the dominant learning theories that have profoundly influencededucation, helping to form the dominant model and principlesmentioned above, have taken little account of imagination.

(Egan, webref)

Imagery: Grayson Wheatley

Grayson Wheatley (b. USA) is a mathematician and mathematics educatorwho has focused particularly on the role of mental imagery in teaching andlearning mathematics.

All meaningful mathematics learning is imaged-based. While there may becertain forms of mathematical reasoning that seem not to use imagery,most mathematical activity has a spatial component. If school mathe-matics is procedural, students may fail to develop their capacity to formmental images of mathematical patterns and relationships. It is well docu-mented that students who reason from images tend to be powerful mathe-matics students. Further, we know that the ability to use images effectivelyin doing mathematics can be developed. When students are encouragedto develop mental images and use those images in mathematics, theyshow surprising growth. All students can learn to use images effectively.Thus, developing spatial sense should be a priority in school mathematics.

(Wheatley, webref)

Wheatley advocates use of tasks such as Quick Draw in which a geometricalfigure is displayed for a few seconds and the learners are challenged to try tomake a quick drawing.


Every child has the power to imagine what is not physically present, and to manip-ulate those images mentally. They also have the power to express those images(though it takes some time before they realise that those images are not sharedwith the people to whom they are talking). Expression takes the form of verballanguage (demands, wishes, instructions, descriptions, chat), but also drawings,movement etc.. There is a real tension between inviting self-expression, andcontrolling the expressions of thirty or more children in a confined space.

Generalising and abstracting

The young child quickly associates different voice tones with the same adult.This is an example of generalisation. So is language. Nouns are labels forclasses of objects, rarely for specific objects (personal names and names forpets and toys being an exception). Thus chair, cup and spoon refer not tospecifics but to general classes. The notion of a noun is both a generalisationand an abstraction.

While people deal with generalities and abstractions all the time, generali-sations play a central role in mathematics, where they are expressed in asuccinct notation which is manipulated so as to draw out further conclusionswhich may be particular or general. Mathematics deals with relationships perse, and so context is of the least importance; hence the prevalence of abstrac-tions in mathematics.

Generalisation: Alfred North Whitehead

As mathematician and philosopher, Whitehead was very concerned abouteducation, taking the view that:

… The progress of science consists in observing … interconnexions andin showing with a patient ingenuity that the events of this ever-shiftingworld are but examples of a few general connexions or relations calledlaws. To see what is general in what is particular and what is permanentin what is transitory is the aim of scientific thought. …

Now let us think of the sort of laws which we want in ordercompletely to realize this scientific ideal. Our knowledge of the partic-ular facts of the world around us is gained from our sensations.

[ … ]… when we have put aside our immediate sensations, the most service-able part – from its clearness, definiteness, and universality – of what isleft is composed of our general ideas of the abstract formal properties ofthings; in fact, … abstract mathematical ideas … . Thus mathematicalideas, because they are abstract, supply just what is wanted for a scien-tific description of the course of events.

(Whitehead, 1911, pp. 4–5)

… what the mathematician is seeking is Generality. … Any limitationwhatsoever on the generality of theorems, or of proofs, or of interpreta-tion is abhorrent to the mathematical instinct.

(ibid., p. 57)

A related and extended version is ‘seeing the general through the particular,and the particular in the general’ (see p. 138). Generalisation is certainlypresent when through examination of a number of cases, often sequential in


some manner, a common pattern is detected, as in recognising the sequenceof odd numbers, or two more than a perfect square. But generalisation oftentakes place on the contemplation of a single example, as David Hilbertdemonstrates.

Generalisation: David Hilbert

David Hilbert (b. Prussia, 1862–1943) was one of the leading mathematicians atthe end of the nineteenth century, making contributions to a wide range ofmathematical topics. His list of 23 problems posed to mathematicians at thebeginning of the twentieth century directed mathematical development forseveral generations of mathematicians. He was associated with a formalist viewof mathematics in which mathematics is seen as the manipulation of formalsymbols according to specified rules. Here is a report from a colleague of his inGöttingen, Richard Courant (1888–1972), who moved to the USA and foundedwhat is now known as the Courant Institute of Mathematical Sciences.

He [Hilbert] was a most concrete, intuitive mathematician who invented,and very consciously used, a principle; namely, if you want to solve aproblem first strip the problem of everything that is not essential.Simplify it, specialize it as much as you can without sacrificing its core.Thus it becomes simple, as simple as can be made, without losing any ofits punch, and then you solve it. The generalization is a triviality whichyou don’t have to pay much attention to. This principle of Hilbert’sproved extremely useful for him and also for others who learned it fromhim. Unfortunately, it has been forgotten.

(Courant, 1981, p. 161)

Both approaches to generalisation, which are sometimes distinguished asempirical and structural or as empirical and generic (see Bills and Rowland,1999) are directed towards detecting and expressing underlying structure.See Krutetskii (p. 139) for further types of generalisation.

For some people, abstraction and generalisation are barely distinguish-able, if at all, while for others, abstraction and generalisation are quitedifferent. While both involve stressing some features and so ignoring(hence removing) other features, which become mere context, generalisa-tion expresses structure, while abstraction is the process of taking thatstructure as the object of study axiomatically (see van Hiele phases, p. 59and structure of attention, p. 63). Thus ‘the sum of two odd numbers iseven, of two evens is even, and of one odd and one even is odd’ expressesa generality within numbers, whereas the structure of ‘the sum of two ofone type is of the second type, two of the second type is of the second type,and of one of each type is of the first type’ abstracts the structure. An alter-native view is to see this as further generalisation by ignoring the context ofodd and even numbers.


Abstraction: Caleb Gattegno

Gattegno sees abstraction as essential, and algebra as vital to humanfunctioning:

Nobody has ever been able to reach the concrete. The concrete is so‘abstract’ that nobody can reach it. We can only function because ofabstraction. Abstraction makes life easy, makes it possible. Words,language have been created by man, so that it does not matter what anyreader evokes in his mind when he sees the word red, so long as whenwe are confronted with a situation we shall agree that we are using thesame word even for different impressions. Language is convenientlyvague so that the word car, for example, could cover all cars, not justone. So anyone who has learned to speak, demonstrates that he can useclasses, concepts. There are no words without concepts. …

Therefore, how can we deny that children are already the masters ofabstraction, specifically the algebra of classes, as soon as they useconcepts, as soon as they use language, and that they of course bring thismastery and the algebra of classes with them when they come to school.

[ … ]… The essential point is this: the algebra is an attribute, a fundamental

power, of the mind. Not of mathematics only.Without algebra we would be dead, or if we have survived so far, it is

partly thanks to algebra – to our understanding of classes, transforma-tions, and the rest. …

(Gattegno, 1970, pp. 23–5)

Abstraction: Richard Skemp

Skemp continues his previous extract (p. 123) on babies having the power toclassify, to extend it into abstraction. There are similarities here with vanHiele phases (see p. 59) and structure of attention (see p. 60).

… [from] successive past experiences of the same object, say a particularchair … we abstract certain common properties … .

We progress rapidly to further abstractions. From particular chairs …we abstract further invariant properties, by which we recognize … anew object seen for the first time … as a member of this class. It is thesecond-order abstraction … to which we give the name ‘chair’. Theinvariant properties which characterize it are already becoming morefunctional and less perceptual – that is, less attached to the physicalproperties of [ the object]. …

From the abstraction chair, together with other abstractions such asthe table, carpet, bureau, a further abstraction, furniture, can be made,and so on. These classifications are by no means fixed. …


Naming an object classifies it. This can be an advantage or a disadvantage.A very important kind of classification is by function; and once an object isthus classified, we know how to behave in relation to it. … But once it is clas-sified in a particular way, we are less open to other classifications.

(Skemp, 1971, pp. 20–1)

Generalisation: Augustus de Morgan

Here de Morgan ponders the need for individuals to participate in abstrac-tion rather than being presented with predigested abstractions.

We now come to a rule which presents more peculiar difficulties in pointof principle than any at which we have yet arrived. If we could at oncetake the most general view of numbers, and give the beginner theextended notions which he may afterwards attain, the mathematics wouldpresent comparatively few impediments. But the Constitution of ourminds will not permit this. It is by collecting facts and principles, one byone, and thus only, that we arrive at what are called general notions; andwe afterwards make comparisons of the facts which we have acquiredand discover analogies and resemblances which, while they bindtogether the fabric of our knowledge, point out methods of increasing itsextent and beauty. In the limited view which we first take of the opera-tions which we are performing, the names which we give are necessarilyconfined and partial; but when, after additional study and reflection, werecur advice to our former notions, we soon discover processes soresembling one another, and different rules so linked together, that wefeel it would destroy the symmetry of our language if we were to callthem by different names. We are then induced to extend the meaning ofour terms, so as to make two rules into one. Also, suppose that when wehave discovered and applied a rule and given the process which it teachesa particular name, we find that this process is only a part of one moregeneral, which applies to all cases contained in the first, and to othersbesides. We have all the alternative of inventing a new name, orextending the meaning of the former one so as to merge the particularprocess in the more general one of which is a part.

(de Morgan, 1898, pp. 33–4)

Note the importance placed on labels and on the accumulation of numerousexamples: other authors recognise the need for structural generalisation(see p. 133 and p. 139).

Generalisation: Lev Vygotsky

Vygotsky emphasises, perhaps more clearly than most, that generalisation isnot a one-off event but a constant and ongoing process:


At any age, a concept embodied in a word represents an act of general-ization. But word meanings evolve. When a new word has been learnedby the child, its development is barely starting; the word at first is ageneralization of the most primitive type; as the child’s intellectdevelops, it is replaced by generalizations of a higher and higher type – aprocess that leads in the end to the formation of true concepts.

(Vygotsky, 1965, p. 83)

Generalisation and symbols: Zoltan Dienes

Zoltan Dienes (b. Hungary, 1916–) went to school in Hungary and Francebefore moving to England where as a teacher, he developed his notion thatyoung children can be taught abstract mathematical structures throughparticipation in games. He is best known for designing and using Multi-baseArithmetic Blocks, also known as Dienes Blocks, for teaching place-valueaddition and subtraction. Here he considers the introduction of symbols forgeneralisations:

The fundamental problem is whether generalization should take placesimultaneously on a broad front, or on several narrow fronts followed byabstraction into a broader front later. Younger children seem to findgeneralizations on a narrow front, that is within certain well-definedfields, easier. This may be part of the developmental pattern. …

One of the problems about using symbolism is how to find the besttime for introducing it. If this is done too early, it tends to be an emptyshell. Classroom work in mathematics can so easily degenerate intolearning certain rules by which the signs can be manipulated, andstudying situations in which they are applicable, each application beingseparately learned. This of course is necessary if the signs do notsymbolize anything. On the other hand, it is possible to wait too longbefore introducing symbolism. When a child has become familiar with amathematical structure he needs a language in which to talk about it,think about it, and eventually transform it. New constructions need newnames, their properties must be described by new symbols if more of thedetail of the structure is to be grasped at one time, and so reflected uponmore effectively. …

There is also the question of whether symbolism can be used as a toolfor cutting through relevant noise during the abstraction process, orwhether it can only be used to formulate what has already beenabstracted. If the latter, then every learning situation will have a ceilingdetermined by (1) the amount of noise generated, (2) the amount ofnoise the learner is able to cut through. …

(Dienes, 1963, pp. 160–1)


See also Bruner (p. 108) on the need for questioning to move beyond theasking of particulars about some generalization stated by the teacher.

Generalisation: Jerome Bruner

Generalisation (see p. 138), or ‘seeing the general through the particular’ is anexample of classification, something that young children quickly learn to do,since it is the basis for language (use of an appropriate noun signals recogni-tion of an appropriate classification of objects as judged by listeners). Involun-tary classification can block as easily as facilitate; intentional classification canbe questioned, interrogated, and the boundaries and uses explored.

Generalising: John Mason

Generalization is the heartbeat of mathematics, and appears in manyforms. If teachers are unaware of its presence, and are not in the habit ofgetting students to work at expressing their own generalizations, thenmathematical thinking is not taking place.

(Mason, 1996, p. 65)

Generalising and specialising

Generalising is only one side of the coin. In order that language can functionthe power to particularise, also referred to as specialising, is vital. Wheneverwe encounter a generality, we check it against our experience, againstparticular cases with which we are familiar. The following extracts illustrateseeing the general through the particular.

Generalising through specialising: Paul Halmos

Another … idea … is to concentrate attention on the definite, the concrete,the specific. … We all seem to have an innate ability to generalize … . Theteacher’s function is to call attention to a concrete special case that hides(and, we hope, ultimately reveals) the germ of the conceptual difficulty.

(Halmos, 1994, p. 852)

There are similarities with seeing the general through the particular (seep. 132).


Generalising and abstraction are the foundations of language, and hence some-thing learners have already demonstrated in using language effectively. A lessonwithout the opportunity for learners to generalise is not a mathematics lesson.

Specialising and generalising: John Mason

Whenever I encounter a generality, I find myself testing it against partic-ular cases. If I am trying to decide whether the general assertion is true,or when it is true, I consider special, often extreme cases. The purpose oftrying out particular cases is not just to seek a counter-example, but toattend to how the calculations are done, with an eye to seeing if theygeneralise. This is exactly what students are expected to do whenlearning a new technique: do some exercises in order to see how thetechnique works [in general] (not just to ‘get the answers’). Specialising isan act I can perform in order to make sense of what always happens, inorder to appreciate and reconstruct generality.

[ … ]There are two important perceptions here:

• seeing the particular in the general (seeing not just a general asser-tion but the opportunity to try out specific particular cases and beingaware of what constitutes particular instances of the general);

• seeing the general through the particular (seeing specific numbersor other aspects as placeholders for other possibilities).

When you find yourself ‘doing an example’ in front of students, youare probably seeing through the particular numbers, the particularcomputations, and are aware of generality. But some numbers may bestructural rather than particular. What are you doing to draw students’attention to the difference?

(Mason, 2002a, p. 108)

Learning is generalising and specialising: Zhoubi Suanjing

Evidence that generalising has been recognised and valued for a very longtime is provided by this quotation from the earliest known mathematics trea-tise in Chinese, the Zhoubi Suanjing written possibly in the first century BC.Here is the conclusion of a dialogue involving master Chén:

… ‘man has a wisdom of analogy’ that is to say, after understanding aparticular line of argument one can infer various kinds of similarreasoning, or in other words, by asking one question one can reach tenthousand things. When one can draw inferences about other cases fromone instance and one is able to generalize, then one can say that onereally knows how to calculate. The method of calculation is therefore asort of wisdom in learning … The method of learning: after you havelearnt something, beware that what you have learnt is not wide and afteryou have learnt widely, beware that you have not specialized enough.After specializing you should worry lest you do not have the ability togeneralize. So by having people learn similar things and observe similar


situations one can find out who is intelligent and who is not. To be ableto deduce and then to generalize, that is the mark of an intelligent man… If you cannot generalize you have not learnt well enough. …

(Li and Dù, 1987, p. 28)

Types of generalisation: Vadim Krutetskii

Krutetskii’s research into mathematical ability led him to a large number ofconclusions. At the core he identified the ability (as seen through Sovietpsychology) to generalise, not just from several examples, but ‘on the spot’:

1 The method of gradual generalization is not the only way to a masteryof knowledge about mathematics; there is another way that differsfrom it in principle but that leads to the same result. Along with themethod of gradual generalization of mathematical material on thebasis of variations in a diversity of particular cases (the method formost pupils), there is another way, in which able pupils, withoutcomparing the ‘similar’, without special exercises or hints from theteacher, independently generalize mathematical objects, relations, andoperations ‘on the spot’, on the basis of an analysis of just onephenomenon, into a number of similar phenomena. They recognizeevery specific problem at once as the representative of a class of prob-lems of a single type and solve it in a general form – that is, they workout a general method (an algorithm) for solving problems of the giventype.

2 Capable pupils generalize mathematical material not only rapidly butbroadly. They very easily find the essential and the general in theparticular, the hidden generality in what seemed to be different math-ematical expressions and problems.

(Krutetskii, 1976, p. 261–2)

Note the similarity with same and different (see p. 128), and with structuralgeneralisation (see p. 133), and with Whitehead (general in particular, p. 132).

Conjecturing and convincing

Making an assertion about a pattern detected is one thing. Justifying it so thatothers are convinced is quite another. Learners learn to make distinctions, todiscriminate in ways which adults find useful, and learn how to justify thosedistinctions and associated conjectures. Young children already displaythese powers in rudimentary forms. Various academic disciplines have moredemanding requirements for justification and convincing, and these areexperienced and developed in school.


Conjecturing: George Polya

George Polya (b. Hungary, 1887–1985) builds on specialising and general-ising as an ascent and descent, in an ongoing process of conjecturing:

… [an inductive attitude] requires a ready ascent from observations togeneralizations, and a ready descent from the highest generalizations tothe most concrete observations. …

First, we should be ready to revise any one of our beliefs.Second, we should change a belief when there is a compelling reason

to change it.Third, we should not change a belief wantonly, without some good

reason.[ … ]The first point needs ‘intellectual courage’. …The second point needs ‘intellectual honesty’. To stick to my conjec-

ture that has been clearly contradicted by experience just because it ismy conjecture would be dishonest.

The third point needs ‘wise restraint’. … ‘Do not believe anything, butquestion only what is worth questioning’.

(Polya, 1954, pp. 7–8)

Conjecturing: Magdalene Lampert

Lampert is unusual in being a mathematics teacher who researches her ownpractice and also educates teachers. Here her work is being described:

… Drawing particularly on Lakatos (1976) and Polya (1954), [Lampert,(1990)] argues that the discourse of mathematicians is characterized by azig-zag from conjectures to an examination of premises through the useof counter-examples. One of her primary goals is to investigate whetherit is possible for students to engage in mathematical activity congruentwith this portrayal of disciplinary discourse. As a consequence, her focusis, for the most part, on how the teacher and students interact as they talkabout and do mathematics. …

(quoted in Cobb, McClain and Whitenack, 1997, p. 273)

Conjecturing and intuition: Efriam Fischbein

Fischbein sees conjectures as expressions of intuitions:

Anticipatory intuitions are conjectures associated with the feeling of totalconfidence. … If the student finds that his conjectures may bemisleading to such a high degree, he may not be willing to make any


more conjectures (at all) or at least to express them publicly (in the class-room). Such an effect would certainly block the student’s solvingcapacity. The student has then to learn to accept the risk of erroneousguesses (even publicly). He should understand that this is the way inwhich everybody solves problems – not only the novice. Certainly we donot consider wild guesses, but only plausible conjectures based onserious preliminary analyses. On the other hand, one has to develop thestudent’s capacity to analyze and check his findings, his anticipatorysolutions, both formally and intuitively. Our belief is that it is possible todevelop, through adequate training, the student’s intuitive feelings ofincongruences, of incompleteness of arguments, of flaws in the lines ofthought. This together with the capacity to analyze formally and system-atically the preliminary solution (the anticipatory intuition) represents anessential condition of success in a problem solving endeavor.

(Fischbein, 1987, pp. 210–11; see also p. 63)

Conjecturing atmosphere: John Mason

Conjecturing provides the fodder for reasoning, for justifying conjectures.Once a conjecture is made, it needs to be challenged, tested, and possiblymodified. Most powerful is to consider different possible conjectures at eachstage (see also Crowley, 1987, p. 9). Often conjectures turn out to be false, sothey need adjusting.

In a conjecturing atmosphere, everything that is said is said because byexpressing it, by getting it outside of yourself, you make it possible tostand back from it and to test it and reason about it and with it. If you tryto keep all your conjectures in your head, they will end up tumblingaround like clothes in a dryer, getting tangled up. In a conjecturingatmosphere, other people invite you to modify your conjecture, or toconsider a particular example that might challenge your conjecture. Youdo not declare someone else to be ‘wrong’; you invite them to ‘modifytheir conjecture’. In a conjecturing atmosphere you take opportunities tostruggle to express when you are unsure, and you take opportunities tolisten to others and to suggest modifications or amplifications or chal-lenging examples, when you are sure.

(Mason, 2002a, p. 109)

Mathematicians rarely solve the initial problems they set themselves. Mostoften they specialise, they conjecture, they modify and remodify until theyfind a problem they can do! When then they leave off work they note one ormore conjectures and some supporting evidence. This is entirely respectable


and sensible. It is much better to summarise work to date and then let go of itthan simply to slink away with a feeling of failure.


Participating in a conjecturing atmosphere in which everyone is encouraged toconstruct extreme and paradigmatic examples, and to try to find counter-exam-ples (through exploring previously unnoticed dimensions-of-possible-varia-tion) involves learners in thinking and constructing actively. This involveslearners in, for example, generalising and specialising.

6 Learning as actionLearning as action

Introduction

From the most ancient of recorded times, educators have stressed the impor-tance of learners being active, as the extract from Plato on page 34 indicates.In the following extract the authors are concerned about the impact of lackof activity.

Enacting not receiving: Banwell, Tahta and Saunders

In general, mathematics cannot be received; it has to be enacted.Commencing in the cradle, it is a symbolizing activity and carries an

emotional charge that can easily explode.It can become a mindless mind training and a refuge from reality or –

unfortunately all too easily – a weapon of subjection and a focus for fear.(Banwell et al., 1972, p. 61)

Describing teaching: Hans Freudenthal

‘Problem-Solving’ and ‘Discovery Learning’ have become catchwords. Inever liked them as mere slogans, and I like them even less since [the]first time I saw them exemplified. Problem solving: solving the teachers’or the textbook authors’ or the researchers’ problems according tomethods they had in mind, rather than the learner himself graspingsomething as a problem. Discovery learning: i.e., uncovering what wascovered up by someone else – hidden Easter eggs. …

… mathematics in individual lives starts [as an activity]. But is thelearner allowed to continue like this? Curious children will not ask forpermission; indifferent and lazy ones prefer to be guided. So in order toexplain how I imagined mathematics would be learned I long ago chosethe term ‘guided reinvention’. It didn’t catch. Should I have done other-wise? Fortunately I did not.


Types of actions: Eric Love and John Mason

The role of the teacher is to stimulate and prompt mathematically relevantactions by learners. In a monograph written for Open University students,Love and Mason began a discussion of modes of teacher interaction bylisting a number of teacher actions designed to promote learner actions:

Messages about mathematics, about its role and value in society andabout how it is learned, are implicit in the ways that actions are carriedout. And these are influenced by views of mathematics itself. Mathe-matics can be seen as a body of knowledge to be handed over to thenext generation (from which comes the metaphor of delivering thecurriculum (like some mail-order package) and it can be seen as aprocess of disciplined inquiry and analysis (from which comes the meta-phor of exploring and investigating). Both views (and there are manyvariants) have an element of fit but when one view is elevated to theview of mathematics it leads to imbalance and impoverishment forpupils. Having an active, exploratory, investigative approach to mathe-matics does not guarantee that pupils will take pleasure in explorationand gain confidence with mathematics, just as an approach based on‘exposition-example-practice’ does not guarantee that all pupils willsucceed or even that all will struggle constructively.

Mathematics teaching involves taking actions in classrooms in order toprompt pupils to act on:

• physical objects by manipulating them (Dienes blocks, attributeblocks, Cuisenaire or colour factor rods, counters, shells and beads,compasses, protractors, rulers, calculators, and so on);

• screen objects (images on computer screens, television, posters);• symbols (numerals, variables, labels, words);• mental objects (pictures-in-the-head, vague senses, intuitions).

Often the action is precipitated by getting pupils to express theirthinking to themselves, to other pupils, to the teacher, and to examiners,via talking, writing, drawing, computer programs, audio and videotapes,and drama.

(Love and Mason, 1992, pp. 1–2)

If learners are to be encouraged to be active, it is important to work out inwhat way they can usefully be active. Doing lots of repetitive exercises is oneform of being active, constructing mathematical objects meeting specifiedconstraints is another, and there are many other possibilities. For example,Lichtenberg said ‘What you have been obliged to discover by yourself leavesa path in your mind which you can use again when the need arises’(Lichtenberg: Aphorismen, quoted in Polya, 1962, p. 99).


On the basis of assumptions and theories, whether implicit or explicit, it ishelpful to consider the types of actions which learners need to engage in, inorder to learn efficiently and effectively.

No matter how tempting it is to blame learners, their conditions, or theteacher, for poor performance, many people have asserted that learners actuallypossess the requisite powers to think mathematically, as the next section shows.

Learner action

What is the basis and nature of learner actions in trying to learn, in trying tomake sense? Are actions purely internal psychological processes? Are they aninternalisation of something experienced in the social environment? Are theypurely an external social act of adopting behavioural practices?

As a reminder that the issue of stimulating learners to be active has deephistorical roots, the next extract is from 1840:

If a child be requested to divide a number of apples among a certainnumber of persons, he will contrive a way to do it, and will tell howmany each must have. The method which children take to do thesethings, though always correct, is not always the most expeditious … Tosucceed it is necessary rather to furnish occasions for them to exercisetheir own skill in performing examples rather than to give them rules.They should be allowed to pursue their own method first, and thenshould be made to observe and explain it; and if it were not the best,some improvement should be suggested.

Examples of any kind upon practical numbers are of very little use,until the learner has discovered the principle from practical examples.When the pupil learns by means of abstract examples, it very seldomhappens that he understands a practical example the better for it;because he does not discover the connexion until he has performedseveral practical examples, and begins to generalise them. (IntellectualArithmetic … , 1840, p. iv)

(McIntosh, 1977, p. 93–4 (reprinted in Floyd, 1981))

So the purpose of learners’ actions is to internalise, comprehend, make senseof, reconstruct abstractions, in response to disturbances experienced (seep. 55, p. 101 and p. 161), through making use of their powers. What thenwould it mean for learners not to be active?

Learners always do something!: Margaret Brown

Margaret Brown (b. Merseyside, 1943–) has spent her career investigatinglearners’ experience of mathematics classrooms, issues in teaching mathe-matics, and issues in supporting teachers. As a result of one of her manygovernment funded projects, she commented:

Learning as action 145

One general point we noticed throughout the interviewing … was thatalmost all children could produce successful strategies for solving prob-lems, even when they did not recognize the operations involved.

(Brown and Küchemann, 1976, p.16 (quoted in Floyd, 1981, p. 9))(See also Brown and Küchemann, 1977, 1981.)

What they did not always do was communicate their method or their resolu-tion effectively on tests. Learners always make sense of what is happening(see van Lehn, p. 210), even if that sense is ‘I can’t’, or ‘Mathematics isn’t forme’, or some other reinforcement of a personal narrative (see Bruner, p. 68and Dweck, p. 112).

Van Lehn and Seeley Brown adopted the approach that learners makesense by adapting techniques in the face of difficulties, gluing together frag-ments of techniques to help them deal with awkward and unexpected situa-tions (see p. 210).

Active learning principles: George Polya

As a highly respected and prolific mathematician, Polya was also highlyrespected as a teacher and as a teacher of teachers. His principal concernwas to teach problem solving (see p. 187). Here he reflects on his teachingprinciples.

Any efficient teaching device must be correlated somehow with thenature of the learning process. We do not know too much about thelearning process, but even a rough outline of some of its more obviousfeatures may shed some welcome light upon the tricks of our trade. Letme state such a rough outline in the form of three ‘principles’ of learning.Their formulation and combination is of my choice, but the ‘principles’themselves are by no means new; they have been stated and restated invarious forms, they are derived from the experience of the ages,endorsed by the judgment of great men, and also suggested by thepsychological study of learning.

[ … ]

1 Active learning. It is been said by many people in many ways thatlearning should be active, not merely passive or receptive; merely byreading books or listening to lectures or looking at moving pictureswithout adding some action of your own mind you can hardly learnanything and certainly you can not learn much.

[ … ]2 Best motivation. Learning should be active, we have said. Yet the

learner will not act if he has no motive to act. He must be induced toact by some stimulus, by the hope of some reward, for instance. Theinterest of the material to be learned should be the best stimulus to


learning and the pleasure of intensive mental activity should be thebest reward for such activity. Yet, where we cannot obtain the bestwe should try to get the second best, or the third best, and lessintrinsic motives for learning should not be forgotten.

For efficient learning, the learner should be interested in the mate-rial to be learned and find pleasure in the activity of learning. Yet,beside these best motives for learning, there are other motives too,some of them desirable. (Punishment for not learning may be the leastdesirable motive.)

Let us call this statement the principle of best motivation.3 Consecutive phases. Let us start from an often quoted sentence of

Kant: Thus all human condition begins with intuitions, proceeds fromthence to conceptions, and ends with ideas. …

[My version of what Kant was saying is] learning begins with actionand perception, proceeds from thence to words and concepts, andshould end in desirable mental habits.

To begin with, please, take the terms of this sentence in some sensethat you can illustrate concretely on the basis of your own experience.(To induce you to think about your personal experience is one of thedesired effects.) …

[ … ]A first exploratory phase is closer to action and perception and moves

on a more intuitive, more heuristic level.A second formalizing phase ascends to a more conceptual level,

introducing terminology, definitions, proofs.The phase of assimilation comes last: there should be an attempt to

perceive the ‘inner ground‘ of things, the material learned should bementally digested, absorbed into the system of knowledge, into thewhole mental outlook of the learner; this phase paves the way to appli-cations on one hand, to higher generalizations on the other.

Let us summarize: For efficient learning, an exploratory phase shouldprecede the phase of verbalization and concept formation and, eventu-ally, the material learned should be merged in, and contribute to, theintegral mental attitude of the learner.

This is the principle of consecutive phases.(Polya, 1962, part II, pp. 102–4)

There are similarities with the See–Experience–Master framework (see p. 263).

Learner as active agent: Herbert Spencer

To tell a child this and to show it the other, is not to teach it how toobserve, but to make it a mere recipient of another’s observations: aproceeding which weakens rather than strengthens its powers of self-


instruction – which deprives it of the pleasures resulting from successfulactivity – which presents this all-attractive knowledge under the aspectof formal tuition – and which thus generates that indifference and evendisgust not unfrequently felt … .

(Spencer, 1878, p. 79)

Learner as active agent: Jerome Bruner

Bruner’s views are summarised as:

To the extent that the materials of education are chosen for their amen-ableness to imaginative transformation and are presented in a light toinvite negotiation and speculation, to that extent education becomes apart of … ‘culture making’. The pupil, in effect, becomes a party to thenegotiatory process by which facts are created and interpreted. Hebecomes at once an agent of knowledge making as well as a recipient ofknowledge transmission.

(Bruner, 1986, p. 127)

This is the essence of what started out as constructivism (learners activelyconstruct meaning) and which then diverged into psychological, radical andsocial constructivism, each with many variants. (See p. 92 for elaboration).

Biological bases

After Spencer, Piaget and Dewey had perhaps the greatest and mostprofound influence on education in the twentieth century. Piaget started as abiologist and became interested in epistemology: the study of how peoplelearn. Since adults are relatively complex, he turned his attention to youngchildren. He was for a time director of a Montessori school in Switzerlandbefore establishing his highly productive and influential institute in Geneva.

Genetic epistemology: Jean Piaget

The fundamental core of Piaget’s conclusions is that human beings activelyconstruct meaning from the situations and environment in which they findthemselves.

Equilibration: Jean Piaget

Piaget saw analogies between an organism responding to changes in itsenvironment, and human beings responding to changes through what wecall learning. A biological way of thinking of this is in terms of equilibra-tion: the organism responds in such a way as to seek equilibrium, thusminimising disturbance.


… I was forced to the conclusion that [the] prime cause was a factor ofgradual equilibration in the sense of auto-regulation. If equilibrium inaction is defined as an active compensation set up by the subject againstexterior disturbances, whether experienced or anticipated, this equilibra-tion will explain, among other things, the more general character calledlogico-mathematical operations – that is, their reversibility (to every direc-tion operation there corresponds an inverse one which cancels it out … ).

(Piaget, 1971, p. 12)

Assimilate–Accommodate: Jean Piaget

Biological equilibration involves assimilating some things and adjusting toaccommodate others. Piaget suggested a cognitive version of these two inter-connected processes.

… when an infant has formed the habit of making objects hung in frontof it swing backward and forward (by pushing them without graspingthem), and when it applies this behavior to some new object that it hasnot seen before, there has been assimilation of this new object or situa-tion into the swinging schema.

(Piaget, 1971, p. 180)

The essential starting point here is the fact that no form of knowledge, noteven perceptual knowledge, constitutes a simple copy of reality, becauseit always includes a process of assimilation to previous structures.

We use the term assimilation in the wide sense of integration intoprevious structures. … without any break of continuity with the formerstate – that is, without being destroyed and simply by adapting … to thenew situation.

… when a baby pulls his blanket toward himself in order to reach someobject that is on it but out of his reach, he is assimilating this situation intoperceptual schemata (the connecting thought is ‘on it’) and active sche-mata (the behavior of the cover on which the object is lying). In short, anytype of knowledge inevitably contains a fundamental factor of assimila-tion which alone gives significance to what is perceived or conceived.

(ibid., pp. 4–5)

This is the famous principle of equilibration, the resolving of disturbance (seep. 55). Piaget uses the example of a liquid in a container: as the container istilted the liquid adapts to the shape of the bottle, but there is no lasting effect.

… adaptation [may be defined] as an equilibrium between assimilationand accommodation … .

… assimilation and accommodation are not two separate functionsbut the two functional poles, set in opposition to each other, of any


adaptation. … There can be no assimilation of anything into theorganism or its functioning without a corresponding accommodationand without such assimilations becoming part of an adaptation context.

… the basic functions of adaptation and assimilation [ … ] embodied inthe most diverse structures, are to be found at every hierarchical level … .

(ibid., p. 173)

Adaptation: Ernst von Glasersfeld

[Piaget saw that] the concept of adaptation could be incorporated in atheory of learning. In my view, this is the major contribution Jean Piagethas made to our understanding of cognition. Eventually this perspectiveled him to the conclusion that the function of intelligence was not, astraditional epistemology held, to provide cognitive organisms with ‘true’representations of an objective environment. Rather, he began to seecognition as generator of intelligent tools that enable organisms toconstruct a relative fit with the world as they experience it.

Though the notion of ‘fit’ was borrowed from the biological concept ofadaptation, it no longer contained the element of preformation or geneticdetermination in the cognitive domain. Here it was the product of intelli-gent construction, of the organism’s own making, as the result of trial,error and the selection of what ‘works’. … Fit or viability in the cognitivedomain is, of course, no longer directly tied to survival but rather to theattainment of goals and the mutual compatibility of constructs.

To make clear and emphasize the instrumental character of knowledge,be it on the level of sensory-motor activities or conceptual operations, Ihave always preferred the term viability. It seems more appropriatebecause, unlike ‘fit’, it does not suggest an approximation to the constraints.

During the last two decades of his life, when Piaget had realized thatthis theory had much in common with the principles formulated bycybernetics, he shifted his focus from the chronology of development inchildren to the more general question of the cognitive organism’s gener-ation and maintenance of equilibrium. In this regard, too, room was leftfor misunderstandings, because the term was not intended to have thesame meaning on all levels of cognition. On the biological/physicallevel, an organism’s equilibrium can be said to consist in its capability toresist and neutralize perturbations caused by the environment. On theconceptual level however, the term refers to the compatibility and non-contradictoriness of conceptual structures.


Assimilate–Accommodate: Richard Skemp

Evidence of the widespread influence of Piaget’s work is found in the usemade of it by many authors, including Skemp, whose thesis was the first in


mathematics education in the UK. His book The Psychology of LearningMathematics (Skemp, 1971) was one of the first directed specifically toteachers and is still widely referred to. He developed a series of primary text-books based on extensive research and development, and his Intelligence,Learning, and Action (Skemp, 1979) describes his conclusions about theways in which learning come about and can be initiated and supported.

A concept can be described as a mental awareness of something incommon among a certain class of experiences. A new concept cannot ingeneral be communicated by a definition. A child needs to be given acarefully chosen collection of examples, from which he or she can makehis own abstraction. These examples may themselves be other concepts,in which case careful planning is required to ensure these lower orderconcepts are available as necessary for the formation of a new one.

The existing structure of knowledge (schema) is thus an essential toolfor further learning. Fitting new ideas into an existing schema is calledassimilation. Sometimes a new idea cannot be thus assimilated withouta modification of the schema, and this is called accommodation. Both ofthese complementary processes are necessary for understanding, andfailure of accommodation is a common cause of difficulty. …

In this context, problems can be seen as tasks which require accommo-dation of the pupil’s existing schema, in greater or lesser degree. They canthus lead to progressive enlargement of the pupil schema. … An under-standing of the process of schematic learning, with its related activities ofassimilation and accommodation, will therefore be a considerable help toteachers who wish to make the best use of problems in their teaching.

Mathematics is concerned with the manipulation of ideas, not of mate-rial objects. This ability to be aware of our own thoughts, and to manipu-late them in various ways, we as adults take for granted. But this ability islargely absent in young children, and only reaches its mature formduring adolescence. It is therefore of great importance for the teachingof mathematics to take account of the various stages of development ofthis reflective use of intelligence, and to seek further knowledge of howwe may help it grow.

(Skemp, 1966, p. 76)

Note the indirect reference to reflective abstraction (see p. 151).

Meaningful learning

Meaningful learning is usually contrasted with rote or meaningless learning,though even if memorised as sounds there is still some meaning for thelearner (sounds to be memorised in sequence for some purpose) even if it isnot the meaning appreciated by the teacher (see also understanding,p. 293). Rote learning has been the bête noire of educators for generations.


Rote learning: Herbert Spencer

The rote-system, like all other systems of its age, made more of the formsand symbols than of the things symbolized. To repeat the wordscorrectly was everything; to understand their meaning, nothing; and thusthe spirit was sacrificed to the letter. … in proportion as there is attentionto the signs, there must be inattention to the things signified … .

(Spencer, 1878, p. 56)

But rote learning is not by itself either harmful or unproductive. It alldepends on whether the fact of having memorised is made further use of, orremains inert (see Whitehead, p. 288; Spencer, p. 35).

Rote learning: Ference Marton and Shirley Booth

Marton and Booth report on studies in Hong Kong with Chinese learnerswho are used to memorising:

Asian learners, it was found, treated memorization as a step towardunderstanding, in that each repetition of a text or lesson give a newperspective on the content, and so an understanding was built up stageby stage. The parallel was drawn with Western actors learning their partsin a new play: Although rote memorization plays a large part in prepara-tion, each repetition, recall, and rehearsal of the lines reveals a new layerof meaning in the part. An amateur actor who merely learns the lines ofthe part, maybe by mnemonic tricks, gives only a wooden representa-tion of the character, whereas a professional who has made the lines hisown by successive repetition and review has thereby extracted thenecessary meaning from the lines and can bring the part to life.


Meaningful and rote learning: David Ausubel

David Ausubel (b. New York, 1918–) was influenced by Piaget. He adopted astrong position concerning the need for learners to have a sense of where theywere being led, which became known as advance organisers (see p. 258).Here he probes the meaning of rote learning (see p. 35), which has botherededucationalists since the earliest recorded reflections on teaching and learning.

… the learner consciously acts upon this concept or [proposition] in anattempt to remember it so that it will be available at some future time. Hemay do this in either of two quite distinct ways. If the learner attempts toretain the idea by relating it to what he knows, and thereby ‘make sense’out of it, then meaningful learning will result. On the other hand, if thelearner merely attempts to memorize the idea, without relating it to his


existing knowledge, then rote learning is said to take place. … anylearning that occurs is not simply either meaningful or rote; it is, insteadmore or less meaningful or more or less a rote … . thus meaningful recep-tion learning will take place when the teacher presents the generaliza-tion in its final form, and the learner relates it to his existing ideas insome sensible fashion. On the other hand, rote reception learning wouldtake place if the teacher presented the generalization, and the studentmerely memorized it. Again, meaningful discovery learning will occur ifthe student formulates meanings the generalization himself and subse-quently relates it in a sensible way to his existing ideas. Finally rotediscovery learning could occur if the learner, having arrived at the gener-alization himself (typically by trial and error), subsequently commit tomemory without relating it to other relevant ideas in his cognitivestructure.

(Ausubel and Robinson, 1969, pp. 44–5)

Actions

In order to avoid the merely rote, learners need to take action.

Actions: Jean Piaget

The foundation of Piaget’s perspective is that human beings engage in actionsin and on the material world, which he saw as a form of research initiatedthrough ‘natural seeking of meaning’. These actions influence and alter theorganism, and mathematics is one means whereby we can study those actions.

The importance of the concept of assimilation is twofold. On the onehand … , it implies meaning, an essential notion because all knowledgehas a bearing on meaning … . On the other hand, this concept expressesthe fundamental fact that any piece of knowledge is connected with anaction and that to know an object or a happening is to make use of it byassimilation into an action schema.

Knowing does not really imply making a copy of reality but, rather,reacting to it and transforming it (either apparently or effectively) in sucha way as to include it functionally in the transformation systems withwhich these acts are linked.

(Piaget, 1971, pp. 5–6)

Mathematics functions for physicists as a language of description, but mathe-matics is much more than that, since it alone can enable us to reconstructreality and to deduce what phenomena are, instead of merely recordingthem. To do this, mathematics uses operations and transformations whichare still actions even though they are carried out mentally.


Mathematics [is], in fact, not simply a system of notations at the service ofphysical knowledge, but an instrument of structuralization, because it isof the nature of operations to produce transformations. The fact that thelatter may be expressible in ‘symbols’ does not in any sense reduce theiractive and constructive nature: thus, the psycho-biological problem ofthe construction of mathematical entities cannot possibly be solved bylinguistic considerations.

(Piaget, 1971, p. 47)

Mathematics consists not only of actual transformations but of allpossible transformations. To speak of transformations is to speak ofactions or operations, the latter being derived from the former, and tospeak of the possible is to speak not simply of linguistic description ofsome ready-made immediate reality but of the assimilation of immediatereality into certain real or virtual actions.

(ibid., p. 6)

So learning is a description of how the organism and its environment influenceand affect each other in a never ending dance, and in Piaget’s view, this appliesat the levels of biology and of the psyche: cognition, affect and behaviour.

Schema: Richard Skemp

Piaget introduced the notion of schema as the structures which peopledevelop which constitute learning, and which direct action and response inthe future. Here Skemp moves from powers to concepts to schemas.

It may be useful to relate some of the terms used so far. … Abstracting isan activity by which we become aware of similarities (in the everyday,not the mathematical, sense) among our experiences. Classifying meanscollecting together our experiences on the basis of these similarities. Anabstraction is some kind of lasting mental change, the result ofabstracting, which enables us to recognize new experiences as havingthe similarities of an already formed class. Briefly, it is something learntwhich enables us to classify; it is the defining property of a class. Todistinguish between abstracting as an activity and an abstraction as itsend-product, we shall hereafter call the latter a concept.

A concept therefore requires for its formation a number of experi-ences which have something in common. Once the concept is formed,we may (retrospectively and prospectively) talk about examples of theconcept.

(Skemp, 1971, p. 22)

There are similarities with Freudenthal concerning concepts (see p. 50 andp. 136) and the process of abstraction (see p. 134).


… many people find it difficult to separate a concept from its name …The distinction between a concept and its name is an essential one … . Aconcept is an idea; the name of a concept is a sound, or a mark on paper,associated with it. …

Naming can also play a useful, sometimes an essential, part in theformation of new concepts. Hearing the same name in connection withdifferent experiences predisposes us to collect them together in ourminds and also increases our chance of abstracting their intrinsic similar-ities (as distinct from the extrinsic one of being called by the samename). Experiment has also shown that associating different names withclasses which are only slightly different in their characteristics helps toclassify later examples correctly, even if the later examples are notnamed. The names help to separate the classes themselves.

(ibid., pp. 23–4)

The criterion for having a concept is not that of being able to say itsname but that of behaving in a way indicative of classifying new dataaccording to the similarities which go to form this concept. …

[ … ]… there are two ways of invoking a concept; that is, of causing it to

start functioning. One is by encountering an example of the concept.The concept then comes into action as our way of classifying thisexample, and our subjective experience is that of recognition. The otheris by hearing, reading or otherwise making conscious the name, or othersymbol, for the concept. … [To do the second requires] the ability toisolate concepts from any of the examples which give rise to them.

[ … ]A concept is a way of processing data which enables the user to bring

past experience usefully to bear on the present situation. Withoutlanguage each individual has to form his own concepts direct from theenvironment. … the concepts of the past, painstakingly abstracted andslowly accumulated by successive generations, become available to helpeach new individual form his own conceptual system.

(ibid., pp. 27–8)

Seeing language as enabling an action in which the past acts upon thepresent has similarities with comments about observation (see p. 31).

The actual construction of a conceptual system is something which eachindividual has to do for himself. But the process can be enormouslyspeeded up if, so to speak, the materials are to hand. It is like the differencebetween building a boat from wood sawn to shape and having to start bywalking to the forest, felling the trees, dragging them home, making planks– having first mined some iron ore and smelted it to make an axe and a saw!

(ibid., p. 29)


… when a number of suitable components are suitably connected, theresulting combination may have properties which it would have beendifficult to predict from a knowledge of the properties of the individualcomponents. …

So it is with concepts and conceptual structures. … a conceptual struc-ture has its own name – schema. The term includes not only thecomplex conceptual structures of mathematics but also relatively simplestructures which coordinate sensory-motor activity. …

Among the new functions which a schema has, beyond the separateproperties of its individual concepts, are the following: it integratesexisting knowledge, it acts as a tool for future learning and it makespossible understanding.

When we recognize something as an example of a concept webecome aware of it at two levels: as itself and as a member of this class.… But this class-concept is linked by our mental schemas with a vastnumber of other concepts, which are available to help us behave adap-tively with respect to the many different situations [we encounter].

(ibid., p. 37)

See also what makes an example exemplary (p. 173). Schemas are essentialfor perceiving and for learning, but may obstruct as well as facilitate.

… if a task is considered in isolation, schematic learning may takelonger. For example, rules for solving a simple equation … can bememorized in much less time than it takes to achieve understanding. Soif all one wants to learn is how to do a particular job, memorizing a set ofrules may be the quickest way. If, however, one wishes to progress, thenthe number of rules to be learnt becomes steadily more burdensomeuntil eventually the task becomes excessive. A schema, even more thana concept, greatly reduces cognitive strain. Moreover, in most mathemat-ical schemas, all the main contributory ideas are of very general applica-tion in mathematics. Time spent in acquiring them is not only ofpsychological value (meaning that present and future learning is easierand more lasting) but of mathematical value (meaning that the ideas arealso of great importance mathematically). In the present context, goodpsychology is good mathematics.

The second disadvantage is more far-reaching. Since new experiencewhich fits into an existing schema is so much better remembered, a schemahas a highly selective effect on our experience. That which does not fit intoit is largely not learnt at all, and what is learnt temporarily is soon forgotten.So, not only are unsuitable schemas a major handicap to our futurelearning: even schemas which have been of real value may cease tobecome so if new experience is encountered, new ideas need to beacquired, which cannot be fitted in to an existing schema. A schema can beas powerful a hindrance as help if it happens to be an unsuitable one.


… a strong tendency emerges towards the self-perpetuation ofexisting schemas. If situations are then encountered for which they arenot adequate, this stability of the schemas becomes an obstacle to adapt-ability. What is then necessary is a change of structure in the schemas;they themselves must adapt. Instead of a stable, growing schema bymeans of which the individual organizes his past experience and assimi-lates new data to itself, [reconstruction is required before the new situa-tion can be understood]. This may be difficult, and if it fails, the newexperience can no longer be successfully interpreted and adaptivebehaviour breaks down – the individual cannot cope.

[ … ]… A schema is of such value to an individual that the resistance to

changing it can be great, and circumstances or individuals imposingpressure to change may be experienced as threats – and responded toaccordingly. Even if it is less than a threat, [reconstruction] can be diffi-cult; whereas assimilation of [a new] experience to an existing schemagives a feeling of mastery and is usually enjoyed.

(ibid., pp. 43–5)

Piaget also took the line that when learners do not understand something, itis the lesson they fail to understand, not the subject matter. Lessons are inef-fective when they attempt to force the learner to move too quickly from qual-itative reasoning and appreciation to quantitative calculations (Piaget, 1973,p. 14). In a later chapter this is recast as taking time to move between doing,talking about what you are doing, and recording in pictures, words andsymbols (see p. 262).

… active intention and consequent practical application of certain oper-ations are one thing, and becoming conscious of them and thusobtaining reflexive and, above all, theoretical knowledge are another.Neither pupils nor teachers suspect that the instruction imparted couldbe supported by all manner of ‘natural’ structures. … speaking to thechild in his own language before imposing on him another ready-madeand over-abstract one, and, above all, in inducing him to rediscover asmuch as he can rather than simply making him listen and repeat.

[ … ]… the methods of the future will have to give more and more scope to

the activity and the groupings of students as well as to the spontaneoushandling of devices intended to confirm or refute the hypothesis theyhave formed to explain a given elementary phenomenon. In otherwords, if there is any area in which active methods will probablybecome imperative in the full sense of the term, it is that in which experi-mental procedures are learned, for an experiment not carried out by theindividual himself with all freedom of initiative is by definition not an


experiment but mere drill with no educational value: the details of thesuccessive steps are not adequately understood.

(Piaget, 1973, pp. 18–20)

The same sentiment could be applied to any task which has pedagogic inten-tions. (See also using situations, p. 249 and p. 250.)

In short, the basic principle of active methods will have to draw its inspi-ration from the history of science and may be expressed as follows: tounderstand is to discover, or reconstruct by rediscovery, and such condi-tions must be complied with if in the future individuals are to be formedwho are capable of production and creativity and not simply repetition.

(ibid., p. 20)

There are similarities with discovery learning (see p. 43 and p. 224).

… since so little learning is retained when it is learned on command, theextent of the program is less important than the quality of the work. Astudent who achieves a certain knowledge through free investigation andspontaneous effort will later be able to retain it; he will have acquired amethodology that can serve him for the rest of his life, which will stimu-late his curiosity without the risk of exhausting it. At the very least, insteadof having his memory take priority over his reasoning power, or subju-gating his mind to exercises imposed from outside, he will learn to makehis reason function by himself and will build his own ideas freely.

(ibid., p. 93)

See also motivation (p. 99).

… [operations] are not constructed nor do they acquire their full struc-tures except through certain exercise that is not verbal alone, but, aboveall and basically, is related to action on objects and on experimentation;properly so called, an operation is an action, but interiorized and coordi-nated to other actions of the same type according to precise structures ofcomposition. On the other hand, these operations are not the attribute ofthe individual alone, but necessarily require collaboration and exchangebetween people. Thus, is it enough for the student to listen for years tolessons, in the same manner as the adult listens to a lecturer, for logic tobe created in the child and adolescent? Or does a real formation of thetools of the intellect require a collective atmosphere of active and experi-mental investigation as well as discussions in common?

(ibid., p. 95)

Note the emphasis on the social as well as on the individual (seeconstructivism, p. 54 and p. 92)


The principal action which learners perform, as emerged in the section onlearners’ powers (see p. 115) is to discern, to make distinctions, to stresssomething as foreground and consequently to ignore something else asbackground. As was previously mentioned, this is considerably developedand extended by Maturana (p. 70) and by the van Hieles (p. 59).

Arbitrary and necessary: Dave Hewitt

Born in England, Hewitt has been a secondary teacher, head of department,and teacher educator. Profoundly influenced by Gattegno, his focus is onlearning (and hence teaching) economically, with a minimum of wastedenergy and time. Here he distinguishes between things you have to be toldas a learner, and things you could find out for yourself. One of the weak-nesses of what came to be labelled discovery methods (see p. 43 and p. 224)was that this distinction was not made. Opponents of discovery recast it asmeaning that children had to rediscover everything for themselves, which ispatently impossible (some things you have to be told) and time-limited (howcan you reconstruct everything developed over thousands of years?).

I start with a proposition: If I’m having to remember … , then I’m notworking on mathematics.

[ … ]… consider how you would respond before reading on.

Student: How many sides has a square got?Teacher: Four.Student: Why?

The only reason why a square has four sides is that a decision wasmade a long time ago to call four-sided shapes with particular properties‘squares’. There is nothing about the shapes which means that they haveto be called squares – indeed, in other languages, the same shapes aregiven different names. Looking at the shapes carefully is not going tohelp students to know what the name of the shapes is, just as looking atthe person does not reveal what their name is. All names, within mathe-matics or elsewhere, are things which students need to be informedabout, and part of a teacher’s role is to inform students of such things.

Once students are informed, there is more work students still need todo. They have to memorise the word and associate the word withshapes with those particular properties. It is typical of the realm ofmemory that not only has a word, for example, to be memorised, butthat word also has to be associated with the right things. Many timesstudents successfully remember a word but may not have made theappropriate association. For example, a student may not call [a


‘diamond’ figure] a square, since the properties they associate withsquare do not include sides which are not horizontal or vertical.

(Hewitt, 1999, p. 2)

I describe something as arbitrary if someone could only come to know itto be true by being informed of it by some external means – whether by ateacher, a book, the Internet, etc.. If something is arbitrary, then it is arbi-trary for all learners, and needs to be memorised to be known. …

It is not only labels, symbols or names which are arbitrary. The mathe-matics curriculum is full of conventions, which are based on choiceswhich have been made at some time in the past. For anyone learningthose conventions today, they may seem arbitrary decisions. Forexample, why is the x coordinate first and y coordinate the second? …

[ … ]There are aspects of the mathematics curriculum where students do

not need to be informed. These are things which students can work outfor themselves and know to be correct. They are parts of the mathe-matics curriculum which are not social conventions but rather are prop-erties which can be worked out from what somebody already knows. …For example, if I turn a quarter turn and then a quarter turn again, I havemade a half turn. It is possible to find out about other fractions of awhole turn without having to be informed. So, the mathematical contentwhich is on a curriculum can be divided up into those things which arearbitrary and those things which are necessary.

(ibid., 1999, pp. 3–4)

If the teacher decides to inform students of some mathematics contentwhich is necessary, then they are treating it as if it is arbitrary, as if it issomething which needs to be told. For example, if a teacher stated thatthe angles inside a triangle add up to half a full turn rather than offering atask for students to become aware of this, then students are left withhaving to accept what the teacher says as true. In this case, it becomesjust another ‘fact’ to be memorised. I call this received wisdom.

(ibid., p. 5)

… an awareness concerning properties can be based upon the adoption ofa convention. For example, having adopted the convention that there are360 degrees in a whole turn and that measurement of turn is based on alinear scale (both arbitrary), then I can use the awareness I have aboutlinearity to say if I halve this, then I halve that. This leads me to be able to saydefinitely that there are 180 degrees in a half a full turn – the certainty workedout through awareness from the original adopted convention.

[ … ]The necessary is about properties, and one possibility is for students

to ‘receive’ properties through the teacher informing them just as for the


arbitrary. However, this turns the necessary into received wisdom andstudents may well treat this as something else to be memorised. Indeed,they will have no other choice unless they are able and willing to do thework necessary to become aware of the necessity of this receivedwisdom. Some students may be able to do this work, in which case thereceived wisdom will become a derived certainty and be known throughawareness rather than memory.

Another choice for a teacher is to provide a task which will makeproperties accessible through awareness. An appropriate task will helpthese properties to be more accessibly known through awareness than ifthe teacher informed students of them and left the students to their owndevices to work out why they must be so.

A teacher taking a stance of deliberately not informing students ofanything which is necessary is aware that developing as a mathematicianis about educating awareness rather than collecting and retaining memo-ries. Furthermore, this stance clarifies for the students the way ofworking which is appropriate for any particular aspect of the curriculum– the arbitrary has to be memorised, but what is necessary is abouteducating their awareness.

If I’m having to remember … , then I’m not working on mathematics.(ibid., pp. 8–9)

Learning phases

Learning cycle: Kurt Lewin and David Kolb

Lewin is best known for his work in the field of organisation behaviour and thestudy of group dynamics. His research discovered that learning is best facilitated


Learning is an active process of exploring, formalising and assimilating.Learners participating in negotiation and creation enhance their learning. Somepeople see action and learning as the same thing.

Learners are constantly equilibrating disturbances perceived in the environ-ment, adapting to their situation through accommodating and assimilating, thatis, internalising, by adjusting old schema or constructing fresh ones. Arrangingthe environment so that it promotes useful, efficient and effective adaptation,challenging, but not excessively, is a fine art.

Learning is usefully seen as more-or-less meaningful and more-or-less rote,rather than simply one or the other. In fact both are tied up together.

The use of powers enables concepts to be formed through the creation andadaptation of schemas. Our ideas and awarenesses are linked together in schemaswhich are triggered by cues in new situations. Learners need to be told what isarbitrary convention, but to have their attention directed to what is necessary as aresult of those arbitrary decisions.

when there is a conflict between immediate concrete experience and detachedanalysis within the individual. His cycle of action, reflection, generalisation, andtesting is characteristic of adult experiential learning (Kolb, 1984).

As long as this cycle is not followed algorithmically, it matches well withexperience of conjecturing in mathematics (see p. 139). Atherton (webref)uses it to structure four aspects of learning styles, two ways of knowing, twoways of understanding through transforming knowledge, and four kinds ofknowledge (see also p. 289 and p. 291).

When probed more deeply, learning seen as action and as reactionchanges its nature over time. Various researchers have sought to capturephases or stages or levels of learning, but most of these have in the endcaused as much confusion and heartache as insight. The problem is thatwhen distinctions are made, they are interpreted as being clear and definite,rather than simply the result of a particular stressing and a particularignoring. So the stages identified by Piagetian researchers must be consid-ered cautiously, since behaviour corresponding to different stages can bewitnessed in the behaviour of learners of very different ages.

Learning stages: Jean Piaget (summarised)

Piaget sought to characterise changes and development in the dominantbehaviour of young learners.

• Sensory-motor attention: pre-verbal, pre-symbolic, spontaneous actionsand reflexes based on acquired habits and acts of enquiry and exploration;

• Pre-operational: words used to denote things and actions; some manipu-lation of symbols or other representations;

• Concrete-operational: logical thought based on experience of physicalmanipulation;

• Formal-operational: hypothetical-deductive reasoning using ideas andsymbols without need for physical manipulation.

Notice the parallels with the structure of attention (see p. 60).Educators pushed Piagetian-based research into establishing age-bands in

which these approaches to the world are supposed to be dominant, but evenas adults it is possible to recognise each of these types of thinking in


Concrete experience

Formation of abstractconcepts and generalisations

Testing implicationsin new situations

Observationsand reflection

ourselves, depending on context and familiarity with a problematic situation.Sometimes we want things to manipulate, which may be labels, symbols,mental images, diagrams or physical objects; sometimes a spontaneous andintuitive act is difficult to articulate or to justify logically. By thinking of theseas forms of attention, of different dispositions triggered in different contexts,it is possible to avoid the ‘stage-age’ mentality which constrained curriculumdevelopment for a generation or two.

There are strong overlaps with another approach to what came to beknown as levels of learning, based originally on secondary school geometry.

Learning phases: van Hiele and van Hiele

Dina van Hiele-Geldof and her husband Pierre van Hiele were students ofFreudenthal, who reported that Dina was able to observe more and moreimportant details in lessons than he could.

When the van Hieles started teaching they were just as unprepared asmany other young teachers; nobody had told them how to do it. Theyhad, of course, passively undergone teaching, and maybe evenobserved how their teachers performed, but this was not enough. Astime went on, they had the opportunity to discuss their own teachingwith each other and with others. They subjected their own actions toreflection. They observed themselves while teaching, recalled what theyhad done, and analysed it. Thinking is continued acting, indeed, butthere are relative levels. The acting at the lower level becomes an objectof analysis at the higher the level. This is what the van Hieles recognisedas a remarkable feature of the learning process, namely in the learningprocess in which they learned teaching. They transferred this feature tothe learning process that was the goal of their teaching, namely thelearning processes of pupils who were learning mathematics. There theydiscovered similar levels. To me this seems an important discovery.

(Freudenthal, 1991, p. 96)

Pierre van Hiele generalised the levels of geometric thinking which he andhis wife identified (see p. 59), distinguishing five phases or stages of learningwhich correspond closely to the structure of attention (see p. 60):

1 In the first stage, that of information, pupils get acquainted with theworking domain. [What Vergnaud calls the conceptual field (see p.199).]

2 In the second stage, that of guided orientation, they are guided bytasks (given by the teacher or by themselves) with different relationsof the network that has to be formed. [Vergnaud might describe thisas developing theorems-in-action; Gattegno might describe this asusing their awarenesses (see p. 61).]


3 In the third stage, that of explicitation, they become conscious of therelations, they try to express them in words, they learn the technicallanguage accompanying the subject matter. [Theorems-in-actionbecome explicit; learners become aware of some of their awarenesses.]

4 In the fourth stage, that of free orientation, they learn by generaltasks to find their own way in the network of relations.

5 In the fifth stage, that of integration, they build an overview of allthey have learned of the subject, of the newly formed network ofrelations now at their disposal. [Through what Piaget called reflectiveabstraction (see p. 171); Gattegno used integration through subordi-nation (see p. 229).]

(van Hiele, 1986, pp. 53–4)

As long as van Hiele’s stages are not seen as definitive of the individual orcompletely distinct, they can be useful. They can serve as a reminder to stim-ulate learners to use their powers. They can also serve as a reminder to allowlearners sufficient time to lay an appropriate foundation so that their learningis efficient and effective, rather than rushing to formal records and algorithmsbefore learners have assimilated and reconstructed the basic actions andprocesses. (See also structure of attention, p. 60.)

Prospective and retrospective learning: Hans Freudenthal

Freudenthal found it valuable to distinguish between prospective and retro-spective learning:

• Prospective learning: taking advantage of predilections and naturaltendencies and past experience; anticipatory or ‘advance organisers’.

• Retrospective learning: recalling old ideas whenever it is apt, andreviewing from a higher stance or in a broader context and from agreater generality.

Intertwining learning strands: integration from past and future learningprocesses. …

… learning should be organised in strands which are mutually inter-twined as early, as long and as strongly as possible. When loose ends areinevitable, they’re taken up at the first opportunity, where they can beconnected to other ones in order to be continued. In a sense, examplesof prospective and retrospective learning can also serve as such for inter-twining learning strands, or at least for points where intertwining canstart in order to be continued more consequently.



There are close similarities with reflection (see p. 280) and especially reflec-tive abstraction (see p. 171).

Turning actions into objects

In the twentieth century, mathematics came to be seen as the study of trans-formations. To study an object you study various actions on that object. Oncea transformation is identified, labelled and denoted by a symbol, it canbecome an object of study upon which to perform further transformations.Looking back over the school curriculum reveals this as an ongoing andcontinuing feature of learning and doing mathematics: processes becomeobjects to be acted upon by further processes.

The important thing about the ways in which actions become objects isthat the actions do not lose their action-quality. Rather a dual perceptiondevelops. Expressions such as 2 – (3 + 4) or 2x

2– (3x + 4) or y = 2x – (3x + 4)

can be seen both as a process (a sequence of operations to be carried out)and as the result of carrying out those actions.

Being central to learning, the process of turning an action or a feeling intoan object, has been studied by many researchers, each using their own vocab-ulary. Using the language of abstraction (see p. 59), Piaget suggested that:

Abstraction comes from action, rather than from the object acted upon.(Piaget, 1972, p. 16)

Indeed, abstraction comes not just not from the action, or even from activeengagement in that action, but from some sort of reflection on or integrationof the experience as not just isolated and one-off, but recurring. Piagetcoined the term reflective abstraction, and this was taken up by Dubinsky(see p. 171) in the context of teaching mathematics. Much of educationalresearch is motivated by desire to learn how such a transformation comes


Piagetian stages are most usefully identified as behaviour rather than as attributesof a an individual: sensory-motor attention; pre-operational; concrete-operational;formal-operational. Van Hiele phases involve thinking in terms of wholes throughvisualisation, analysis, abstraction, informal and formal deductions. Translated intostructure of attention, it is possible to move rapidly between, or even to experi-ence simultaneously, attention on wholes, on distinctions, on relationships, onproperties and perceived objects, and on properties as independent of any partic-ular instance.

Prospective learning through anticipation, and retrospective learning throughrecollection and reflection can be intertwined to produce effective learning.

Any one can be working at many of these at the same time, but apparentlack of competence may indicate some disruption, some as yet unintegrateddistinction making, relationships, properties, etc..

about and how it can be fostered and exploited in the teaching ofmathematics.

Compression: William Thurston

William Thurston (b. USA) is a highly productive and creative researchmathematician.

Mathematics is amazingly compressible: you may struggle a long time,step by step, to work through some process or idea from severalapproaches. But once you really understand it and have the mentalperspective to see it as a whole, there is often a tremendous mentalcompression. You can file it away, recall it quickly and completely whenyou need it, and use it as just one step in some mental process. The insightthat goes with this compression is one of the real joys of mathematics.

(Thurston, 1990, pp. 846–7)

Procepts: Eddie Gray and David Tall

Gray is a mathematics educator who has concentrated on pupils in earlyyears, while his colleague, Tall, has focused largely on learners at tertiarylevel. Nevertheless they find much in common, and in particular, how aprocedure or action carried out can become an object of study, henceprocess-object becomes procept. For example, the symbol-expression 3x – 2can be seen both as specifying a calculation procedure given a value for xincluding an as-yet-unknown value, and also the answer to any such calcula-tion, that is, the generality being expressed. Similarly, 2/3 is both a divisionoperation and the answer to that operation being carried out.

We do not consider that the ambiguity of a symbolism expressing theflexible duality of process and concept can be fully utilized if the distinc-tion between process and concept is maintained at all times. …

An elementary procept is the amalgam of three components: a processthat produces a mathematical object, and a symbol that represents eitherthe process or the object.

(Gray and Tall, 1994, p. 121)

Although their evidence draws only on early arithmetic, Gray and Tall indi-cate that the same proceptual thinking is required at all levels of mathe-matics. They go on to cite research showing that learners show evidence ofqualitatively different approaches which can be accounted for in terms ofprocepts. They found that proceptual thinking is necessary to progress inmathematics, but that some learners find this move more difficult thanothers, creating a proceptual divide.


1 Procedural thinking is characterized by a focus on the procedure andthe physical or quasi-physical aids that support it. The limiting aspectof such thinking is the more blinkered view that the child has of thesymbolism: numbers are used only as concrete entities to be manipu-lated through a counting process. The emphasis on the procedurereduces the focus on the relationship between input and output,often leading to idiosyncratic extensions of the counting procedurethat may not generalize.

2 Proceptual thinking is characterized by the ability to compress stagesin symbol manipulation to the point where symbols are viewed asobjects that can be decomposed and recomposed in flexible ways.

[ … ]… We therefore hypothesize that what might be a continuous

spectrum of performance tends to become a dichotomy in whichthose who begin to fail are consigned to become procedural. Webelieve that this bifurcation of strategy – between flexible use ofnumber as object or process and fixation on procedural counting – isone of the most significant factors in the difference between successand failure. We call it the proceptual divide.

(ibid., p. 132)

Once difficulties are encountered and processes remain as process (comparethis with theorems-in-action, see p. 63), learners fall further and further behind.In their summing up, Gray and Tall call upon Thurston’s use of compression.

… it is the compression of mathematical ideas that makes them sosimple. As proceptual thinking grows in conceptual richness, proceptscan be manipulated as simple symbols at a higher level or opened up toperform computations, to be decomposed and recomposed at will. Suchforms of thinking become entirely unattainable for the proceduralthinker who fails to develop a rich proceptual structure.

(ibid., p. 137)

Proceptual thinking is not a quality of a person, but of a person in a situationat a particular time regarding a particular concept. At any time and at anystage, learners can experience a topic ‘running away from them’ due toprocesses not becoming objects.

Reification: Anna Sfard

Anna Sfard (b. Poland) is an Israeli researcher and educator who uses theterm reification for processes becoming objects. She distinguishes two kindsof definitions and concepts: operational and structural. Operational ones arespecified in terms of actions, whereas structural ones are specified in thelanguage of objects. She notes that similar distinctions have been made using


the language of conceptual entities and reasoning procedures (Harel andKaput, 1992), and dynamic and static interpretations (Goldenberg, Lewisand O’Keefe, 1992).

[There is] a delicate distinction between the terms object and entitywhich seem to be used by other authors as synonymous. …

(Sfard, 1992, p.60)

Entity refers to an integrated whole while object conveys an assumptionabout existence.

… it is important to emphasize that the terms process and object are to beunderstood … as different facets of the same thing rather than as totallydistinct, separate components of the mathematical universe. In otherwords, the operational and structural modes of thinking, although osten-sibly incompatible, are in fact complementary.

[ … ]An important question … concerns the origins of mathematical

objects: where did these abstract entities come from, in both historicaland psychological senses?

(ibid., pp. 60–1)

Sfard (1991) tracks the development of the concept of function which shesees as a three-centuries long struggle for reification. She concludes from herhistorical and observational researches two principles:

1 New concepts should not be introduced in structural forms. …2 A structural conception should not be required as long as the student

can do without it. …(Sfard, 1992, p. 69)

These principles may be cast a little strongly, but they are consistent withPiaget’s notion of action and knowing and learning coming about throughaction on familiar objects. There is no point in throwing learners objects asexamples which are not already familiar and object-like for them. This prin-ciple is expressed slightly differently in the SEM and MGA frameworks(p. 263). Sfard identifies three stages of concept construction: interiorisation,condensation and reification.

At the stage of interiorisation a learner gets acquainted with theprocesses which will eventually give rise to a new concept … . Theseprocesses are operations performed on lower-level mathematicalobjects. Gradually, the learner becomes skilled at performing theseprocesses. … we would say that a process has been interiorised if it ‘canbe carried out through [mental] representations’ (Piaget, 1970, p. 14),


and in order to be considered, analyzed and compared it needs nolonger to be actually performed.

(Sfard, 1991, p. 18)

From an activity theory (see p. 84) perspective, interiorisation is only possibleif the same process is encountered in the behaviour of relative experts.

The phase of condensation is a period of ‘squeezing’ lengthy sequencesof operations into more manageable units. At this stage a person becomesmore and more capable of thinking about a given process as a whole,without feeling an urge to go into details. … This is the point [when aname or label is attached] at which a new concept is ‘officially born’. … Aprogress in condensation would manifest itself also in growing easiness toalternate between different representations of the concept.

[ … ]… Only when a person becomes capable of conceiving the notion as

a fully-fledged object, we shall say that the concept has been reified.Reification, therefore, is defined as an ontological shift – a sudden abilityto see something familiar in a totally new light. Thus, whereasinteriorisation and condensation are gradual, quantitative rather thanqualitative changes, reification is an instantaneous quantum leap: aprocess solidifies into object, into a static structure. Various representa-tions of the concept become semantically unified by this abstract, purelyimaginary construct. The new entity is soon detached from the processwhich produced it and begins to draw its meaning from the fact of itsbeing a member of a certain category. … A person can investigategeneral properties of such [a] category and various relations between itsrepresentatives. He or she can solve problems involving findinginstances of the category which fulfil a given condition.

(ibid., pp. 19–20)

Objects: Willi Dörfler

Willi Dörfler is an Austrian mathematician, mathematics educator andresearcher who combines practicality with philosophical analysis.

… at some point … the learner has to make up his or her mind (a meta-phoric expression) whether to consider ‘this’ (i.e., a matrix, a group, agraph, a term) as an object in its own right. This decision is combinedwith a change in point of view: Instead of regarding the various elementsin their own right and with their individual properties and relations, theyare viewed as forming a whole with distinct properties and relations.There is also a switch of focus of attention from the single elements totheir totality and the emerging qualities of the new entity. … Languagesupplies us with many means to realize linguistically this change of


perspective. It is a change in the way of speaking and thinking about therespective piece within our experience. Of course, there have to bereasons for adopting this new perspective.

… This object is kind of virtual, imagined, or assumed. Of course, afterhaving made the decision for this change of perspective, nothingprevents one from considering the entity as a fully-fledged unit or object.This is supported by investigating the properties of the object as a wholeand its behavior as an entity, which is categorically different from theproperties of the constituting elements. Again, this is a way of describingthe entity as an object and is formed during the process of describing itsbehavior. The object, so to speak, lives simply and exclusively in itsdescriptions and needs no other place to gain existence and relevance.

… Because the object was formed deliberately by an intentional act,it has no permanence beyond how and how long one wants to have itavailable as a unit. One can put it aside for some time and revive it ifappropriate.

(Dörfler, 2002, pp. 342–3)

I do not claim that … change of perspective to objects comes by itself or isalways easy to make. … The awareness that learning is partly an encultur-ation and an initiation into a way of speaking, of personally acceptingspecific decisions, attitudes, points of view, and conventions might changedramatically the overall attitude of students toward mathematics. If the so-called construction of mathematical objects is in fact interpreted as a deci-sion to treat something as a unified or reified entity, then this decision has tobe taken by each learner. The role of the teacher then would be to makethis decision appear as sensible and plausible as possible. The newperspective for the students must be worth trying and they must feelempowered to enter into the proposed view. In principle, this essentiallywould also comprise the possibility of refusing the decision or adopting anew perspective only if it will prove successful and viable. I believe thatmathematical constructions are not automatic for learners; rather, they haveto agree to them, to accept them, and to decide to use them. That is, thelearner has to want to treat something as a unit and an object. In the courseof time, this decision for objectification then becomes a deliberate practiceand a thinking strategy. The switch to an object view (or entity view, or unitview) can then be carried out at will as the need may arise, and, moreimportant, can be reversed with the same ease even in hitherto unfamiliarsituations. A model based on decision making better reflects the highdegree of flexibility in handling mathematical objects (in creating anddissolving them), as shown by experienced students of mathematics.

(ibid., p. 346)

Notice how this extends the van Hiele phases (see p. 59) by suggestingmovement in both directions among them.


Reflective abstraction: Ed Dubinsky

Ed Dubinsky (b. Pennsylvania, 1935–) is a mathematician who has worked hardto promote the cause of mathematics education at university level. He hasdeveloped software to enable undergraduates to manipulate concrete (virtual)objects in group theory, and following on from Piaget, has developed what hecalls APOS theory as an approach to effective teaching (see below). Here hedescribes three forms of abstraction identified by Piaget:

Empirical abstraction derives knowledge from the properties of objects(Beth and Piaget, 1966, pp. 188–189). We interpret this to mean that it hasto do with experiences that appear to the subject to be external. Theknowledge of these properties is, however, internal and is the result ofconstructions made internally by the subject. According to Piaget, thiskind of abstraction leads to the extraction of common properties ofobjects and extensional generalizations, that is, the passage from ‘some’ to‘all’, from the specific to the general (Piaget and Garcia, 1983, p. 299). …

Pseudo-empirical abstraction is intermediate between empirical andreflective abstraction and teases out properties that the actions of thesubject have introduced into objects (Piaget, 1985, pp. 18–19). Consider, forexample the observation of a 1–1 correspondence between two sets ofobjects which the subject has placed in alignment … . Knowledge of thissituation may be considered empirical because it has to do with the objects,but it is their configuration in space and relationships to which this leadsthat are of concern and these are due to the actions of the subject. Again, ofcourse, understanding that there is a 1–1 relation between these two sets isthe result of internal constructions made by the subject.

Finally, reflective abstraction is drawn from what Piaget (1980, pp. 89–97) called the general coordinations of actions and, as such, its source isthe subject and it is completely internal … [for example] seriation, inwhich the child performs several individual actions of forming pairs,triples, etc., and then interiorizes and coordinates the actions to form atotal ordering (Piaget, 1972, pp. 37–38). This kind of abstraction leads toa very different sort of generalization which is constructive and results in‘new syntheses in midst of which particular laws acquire new meaning’(Piaget and Garcia, 1983, p. 299).

(Dubinsky, 1991, p. 97)

… when properly understood, reflective abstraction appears as adescription of the mechanism of the development of intellectualthought. …

[ … ]According to Piaget, the first part of reflective abstraction consists

of drawing properties from mental or physical actions at a particularlevel of thought. … This involves, among other things, cognizance or


consciousness of the actions … . Whatever is thus ‘abstracted’ isprojected onto a higher plane of thought … where other actions arepresent as well as more powerful modes of thought.

It is at this point that the real power of reflective abstraction comes in for,as Piaget observes, one must do more than dissociate properties from thosewhich will be ignored or separate a form from its content … . There is ‘aprocess which will become increasingly evident over time: the constructionof new combinations by a conjunction of abstractions’ (Piaget, 1972, p. 23).

(ibid., p. 99)

APOS Theory: Ed Dubinsky

APOS is short for actions, processes, objects, schemas as a development whenmaking sense of situations and solving problems:

We begin with objects. These encompass the full range of mathematicalobjects: numbers, variables, functions, … each of which must beconstructed by an individual at some point in her or his mathematicaldevelopment.

At any point in time there are a number of actions that a subject canuse for calculating with these objects. These actions go far beyondnumerical calculation resulting in numerical answers. …

It is possible for a subject to work with actions in ways other than justapplying them to objects. First, an action must be interiorized. … thismeans that some internal construction is made relating to the action. Aninteriorized action is a process. Interiorization permits one to be consciousof an action, to reflect on it and to combine it with other actions. …

Interiorizing actions is one way of constructing processes. Anotherway is to work with existing processes to form new ones. This can bedone, for example, by a reversal. … If the process is interiorized, thestudent might be able to reverse it to solve problems. …

[ … ]In addition to using processes to construct new processes, it is also

possible to reflect on a process and to convert it into an object. [This isencapsulation.]

(Dubinsky, 1991, pp. 106–8)

See also Dubinsky (webref) and Davis, Tall and Thomas (webref). There aresimilarities with the Manipulating–Getting-a-sense-of–Articulating frame-work (see p. 264). Transference (see p. 291) is yet another description ofhow ideas and actions become objects, as is the transition of theorems-in-action into theorems (see p. 63).

Actions becoming objects of study with relationships, properties and evenaxioms (see structure of attention, p. 60) are hard enough to trap in yourself.Arranging conditions to foster the same transitions in learners is a challenge.


As soon as you try to ‘make it happen’, it becomes instruction rather thaneducation; procedural content to be memorised rather than reconstructedand reconstructible experience of sense-making on the part of the learner.(See didactic transposition, p. 83 and the constructivist dilemma, p. 96.

Learning from examples

As Isaac Newton said so beautifully in his book on algebra:

… because craft skills are more easily learnt by example than by precept,it seems appropriate to adjoin the solutions of the following problems.

(Whiteside, 1968, p. 429)

Examples are and always have been the mainstay of mathematics teaching.But what are learners supposed to do with examples? What actions do theyneed to perform so that what is exemplary to the teacher about an example,becomes exemplary for the learner? Put another way, if you do not appreciatewhat something is supposed to be an example of, how can it be an example ofanything (relevant)? There is a growing literature on this question alone (seeAnthony, 1994; Atkinson et al., 2000; Benbachir and Zaki, 2001; Bills, 1996; Billsand Rowland, 1999; Chi and Bassok, 1989; Dahlberg and Housman, 1997).

When is an example exemplary: Anna Sierpinska

Anna Sierpinska raises the question of how something becomes exemplary:

It is a pedagogical adage that ‘we learn by examples’. Pedagogues, ofcourse, think of paradigmatic examples in this case. They think ofinstances that can best explain a rule or a method, or a concept.

The learner is also looking for such paradigmatic examples as he orshe is learning something new. The problem is, however, that beforeyou have a grasp of the whole domain of knowledge you are learning,you are unable to tell a paradigmatic example from a non-paradigmaticone. So you make mistakes, wrong choices, wrong generalizations(because, of course, you generalize from your examples). Moreover, asthe example is normally represented in some medium (enactive, iconic


Actions or processes becoming objects (reification, encapsulation, compression) canbe thought of as precepts to reinforce their dual nature. One way to describe how thishappens is as a sequence of interiorisation, condensation, and reification. Anotherdescription is in terms of empirical abstraction, pseudo-empirical abstraction andreflective abstraction. APOS ‘theory’ summarises this movement and has been used toinform teaching and curriculum development at tertiary level. Yet another descriptionis in terms of educating awareness and shifts in the structure of attention.

or symbolic) you may mistake the features of the representation for thefeatures of the notion thus exemplified.


It seems then that ‘learning by examples’ is a property of our minds that haslittle in common with the pedagogical expectations expressed in the adage.An example is always embedded in a rich situation that contains moreelements, data, information than just those directly related to the objectexemplified. The teacher cannot be sure that, from this sea, the studentswill fish only the bits strictly relevant for the formation of the concept. It is[not] hard, therefore, to understand the teacher’s frustration, when, afterhaving prepared the best examples, he or she finds that the students are stillable to commit the most unbelievable mistakes and errors. The method ofparadigmatic examples is not really a method of teaching. Rather, it is a wayin which concepts are being formed: the examples cannot be transmittedfrom the teacher’s mind to the learner’s mind. The latter must construct orreconstruct examples that can be regarded as paradigmatic in some moreobjective sense. The teacher can only help the learner by organizing situa-tions against which the consecutive tentative forms of these examples canbe tested, in which they can be revealed, and in which a change can bediscussed and negotiated, if necessary.

Examples are, in understanding abstract concepts, the indispensableprop and the necessary obstacle. It is on the basis of examples that wemake our first guesses. When we start to probe our guesses, the funda-mental role is slowly taken over by the definitions.

(ibid., p. 91)

There are similarities with van Hiele phases (see p. 59), and with Marton’snotion of dimensions-of-possible-variation (see p. 56).

One of the questions to ask about examples is whether there are ambigu-ities between structural features (which are what make the thing an example)and special features which are particular to the example. For instance, a circleof radius 2 has a circumference of 2 × p × 2 and an area of p × 2

2. What is the

difference between the various twos? Is it evident to learners?

Practice and skills

Finally, we return to the issue of manipulative skill and facility with tech-niques which traditionally is seen as best learned through practice. Certainlyprior to an examination, practice is valuable in getting up speed, which


What is exemplary for one person can be completely particular and isolated forsomeone else. Do learners appreciate what is exemplary: what can change andwhat has to remain invariant?

means reducing the amount of focused attention required to perform thetechnique so that you can be on the look out for slips and wrinkles, andmaintain a direction towards the solution of the problems.

Practice: Mary Boole

Mary Boole uses the notion of ‘fictions’ for pedagogic devices, whetherapparatus, images or symbols, mnemonics or other memory devices, whichare introduced temporarily to aid comprehension but which ultimately haveto be discarded or left behind if learners are to develop facility.

… Truth is never received into the human mind without an admixture ofconventions, of what may be called fictions. These fictions have to beintroduced, used, and then withdrawn. It would be impossible to teacheven so straightforward a subject as mathematics without the temporaryuse of statements which are not true to the nature of things. The historyof a child who is learning mathematics, like that of human thought, isvery much a record of alternate introduction of convenient fictions andsubsequent analysis of their true nature. A class, like the public, tends attimes to become groovy and mechanical; to mistake the accidental forthe essential; to treat necessary aids to learning is if they were actualtruths; to lose sight of the relative importance of various kinds of infor-mation. A class in Botany tends to forget that classification and termi-nology are not so much part of the life of plants as circulation andfertilization; a class in analytical Geometry forgets that the coordinatesare no part of a curve. … A student tends to such forgetfulness inproportion as he becomes mechanical in his work; the genius of ateacher is very much shown by the manner in which he contrives toarouse the interest and correct the errors of a class which is becomingtoo mechanical.

Theorists in education sometimes imagine that a good teacher shouldnot allow the work of his class to become mechanical at all. A year ortwo of practical work in the school (especially with examinationslooming ahead) cures one of all such delusions. Education involves notonly teaching, but also training. Training implies that work shouldbecome mechanical; teaching involves preventing mechanicalness fromreaching the degree fatal to progress. We must therefore allow much ofthe actual work to be done in a mechanical manner, without directconsciousness of its meaning; and intelligent teacher will occasionallyrouse his pupils to full consciousness of what they are doing; and if hecan do so without producing confusion, he may be complemented andhis class congratulated.

(Tahta, 1972, p. 15)


Note links with awareness (see p. 61) and reflection (see p. 280), with pedagogictools and cultural mediation (see p. 85), and with rote learning (see p. 152).

Practice makes perfect: Shiqi Li

Shiqi Li writes from a culture in which memorisation is the foundation stoneof learning. But care is needed when connecting with learning by rote as it isconceived in the West. For as Marton (Marton and Booth, 1997) has found,memorisation is only the starting point for achieving understanding over aperiod of time, just as people often value as adults having had to memorisegood poems, speeches from Shakespeare, and the like, when they wereyoung.

Li neatly summarises much of the writing about how actions become ‘learning’.

It is now widely accepted that teaching mathematics means teachingthrough mathematical activities. Activity or action, in other words,involves thinking, doing and physical or mental manipulation. Why isactivity necessary? Because the form of activity has profound implica-tions and performs particular functions for cognition.

First of all, mathematics learning involves (quasi-) empirical actions(Lakatos, 1976). Manipulative practice is a fundamental action for mathe-matics. No matter how students are to learn, whether in the traditionalway with pencil and paper, or, in a more modern way, with the aid of acomputer, mathematics is not learned by a wild flight of fancy. Althoughpeople sometimes may have a sudden inspiration in problem solving, itis still dependent on [ac]cumulating experience. Students, the same asmathematicians, will do mathematics themselves: knowledge is acquiredthrough practical activities. In some aspects, the practicality of mathe-matics is not completely the same as that of experimental sciences suchas physics, chemistry, etc.. However, behavioral or mental operation isstill needed in mathematical activity.

[ … ]Nevertheless, mathematical concepts do not come directly from fact.

They are not abstracted from actual things, but are a product of coherent


, corresponding to the English proverb practice makes perfect, isan ancient Chinese idiom. Many teachers in China as well as in East Asiabelieve it and consider it to be a general principle for all kinds oflearning: through imitation and practice, again and again, people willbecome highly skilled. Is practice an effective way of teaching andlearning, especially for mathematics education? In fact, I cannot answersimply ‘yes’ or ‘no’ to this question.

(Li, 1999, p. 33)

actions on things. So a mathematical object is actually a particular object,a so-called mental object. For instance, the concept of addition is notproduced from more marbles but from the process of adding orcombining. …

[ … ]What psychological mechanism is needed by students in this course of

concept formation? The basis of the organizing concept is provided byexperience but the concept itself is not provided. To construct aconcept, a more important thing is a leap in thinking, i.e. reflectiveabstraction (Piaget, 1970). …

Without manipulation, the subsequent reflection cannot be put intoeffect. And if there were not a sufficiently strong background of manipu-lation, many contexts and properties would only be viewed as acci-dental phenomena that would not enable students to discover preciseconclusions. Therefore students’ manipulative practice in their learningwill lay a foundation for their reflective abstraction. Moreover, theseactivities must be of their personal experience. Students should beinvolved in practical activities, organizing situations, sending messagesand constructing their understanding. Even to see someone else’s doing,s/he must perform it her/himself and make sense of her/his manipula-tion. Nothing can take the place of her/his own thinking. One of thefunctions of routine practice that we stress is to urge students to partici-pate in their activities and let them learn swimming by swimming. Inshort, practice makes perfect; perfect will be formed on the solid basis ofpractice. If there were no fundamental activities, reflective abstractionwould be a castle in the air.

(ibid., pp. 33–4)

Li quotes Freudenthal:

If a learning process is to be observed, the moments that count are itsdiscontinuities, the jumps in the learning process (Freudenthal, 1978, p. 78).

(ibid., p. 35)

Algorithms drive out thought: Sherman Stein

Sherman Stein (b. California, 1930–) is a mathematician known particularlyas a populariser of mathematics. Here he points out that ‘teaching thinking’can ever so quickly become ‘teaching how to solve classes of problems’which easily turns into teaching learners algorithms so they do not have tothink. He recommends open-field tasks:

What is the point of [an] exercise? Is it to check a definition [is under-stood] or a theorem or the execution of an algorithm? … Blinders are


placed on the student to focus attention on particular facts or skills. Forinstance we may ask students to factor x4 – 1.

An open-field exercise puts no blinders on the student. We might ask‘For which positive integers n does x2 – 1 divide xn – 1?’ An open-fieldexercise may not connect with the section covered that day; it may noteven be related to the course. Such an exercise may require the studentto devise experiments, make a conjecture, and prove it. If it has all threeparts, it is a ‘triex’, which is short for ‘explore, extract, explain’ or for ‘trythe unknown’.

[ … ]… Prove that if 3 divides the sum of the digits of an integer, then 3

divides the integer. …… If 3 divides the sum of the digits of an integer, must it divide the

integer? …… Let d be one of the integers 2 through 9. If d divides the sum of the

digits of an integer, must it divide the integer? …(Stein, 1987, pp. 3–4)

What Stein is suggesting can be cast in terms of stimulating learners toexperience dimensions-of-possible-variation (see p. 56), and in terms ofpsychologising the subject matter (see p. 45 and p. 203) in order to interestlearners by getting them to use their natural powers (see p. 115 and p. 233).

Reversions: Bob Davis

Bob Davis (b. Maryland, 1926–1997) was a mathematics teacher andeducator who was inspired by the teaching of Gattegno and other colleaguesassociated with the Association of Teachers of Mathematics in the UK, andwho in turn inspired generations of teachers in the USA through his MadisonProject. He fashioned and championed an approach to teaching which stim-ulated learners to reconstruct for themselves important mathematical proper-ties, relationships and concepts. Here he reports a colleague noticing someclassic errors: 4 × 4 = 8; 2

3= 6; 6 ÷ 1

2 = 3.

… in every case, the student was giving a correct answer – but a correctanswer to a different question. The student had not answered the orig-inal question …

[Davis’ colleague] proposed, and tested, a remediation procedure. Herecommended that the teacher figure out the question … that thestudent had answered; the teacher should then ask [that question]. [He]predicted that, in nearly every case, the student would not answer [thatquestion] but would immediately correct the answer to [the originalquestion]. …Teacher: how much is seven times seven?Student (grade seven): Fourteen.


Teacher: How much is seven plus seven?Student: Oh! It should be forty-nine!

The frequency of dialogues on this pattern suggests that there is somekind of echoic ‘second hearing’ or … ‘instant replay capability’. … Itreveals something interesting about the student’s control structure, andabout the student’s understanding of the teacher’s goals, that the studentdoes NOT bother answering the second question, assuming (correctly)that what was really wanted was a correct answer to the original question.

(Davis, 1984, pp. 100–1)

Until you actually notice a learner doing a reversal, it remains a ‘theoreticalpossibility’. A label such as reversal can help alert you to the phenomenonwhen it happens, thus opening up the possibility of responding constructively.

Paradigm teaching strategy: Bob Davis

Davis describes his work on a task known as ‘pebbles in a bag’, in which abag with an unknown number of pebbles in it has known numbers ofpebbles added and removed, and learners are invited to develop a notationto enable them to keep track of the number of pebbles in the bag relative tothe number at the start. Davis draws out principles for task design which hecalls paradigm teaching strategy:

First, it involves ideas for which virtually all students have powerful repre-sentations. For example, nearly every student knows that if you have abag that is partially filled with pebbles, and if you then put two pebblesinto it, it will have more pebbles in it than it did before you did that.

Second, it is a reliably accurate ‘isomorphic image’ for all operationsof the form A + B and A – B, where A and B are positive integers … .

[ … ]Finally, it is simple … .[ … ]… All important mathematics ‘makes sense’ – but more often than not

one must have appropriate experiential background in order to under-stand the challenge that is being faced, and the nature of an effectiveresponse. Education must provide the experience, not assume it.

(Davis, 1984, pp. 314–15)

Subordinating: Dave Hewitt

Inspired by Gattegno, Hewitt’s focus is teaching and learning economically,which means working with attention and awareness. In his thesis hedevelops a number of themes, among which is the idea that if you wantsomeone to develop facility, then they need to integrate that functioning,


reducing the amount of attention which is required to act in the specifiedmanner. To achieve this, you divert their attention, attracting them to placemuch of it in some higher-order task:

Principle of economy: place attention in an activity which subordinatesthe desired learning. This means that something is practised whilst prog-ress is made at a higher subordinate level (practice through progress).

(Hewitt, 1994, p. 167)

In order to get learners to practise (integrate) a technique, Hewitt’s practice isto offer learners a task which involves them in using that technique on exam-ples which they construct in order to reach and check conjectures. Noteresonances with Dewey (p. 127 and p. 233).


Practice which diverts attention away from the details of doing, so that theactions are integrated into automaticity, is helpful. Practice which meanspaying as little attention as possible but ‘getting the answers’ is not (comparewith the didactic tension).

7 Learning what?Learning what?

Introduction

Making lists of mathematical definitions, topics and techniques does notreally answer the question of what it is that we want people to learn fromyears of school mathematics. Few of us solve quadratic equations or factorisequadratic expressions while shopping or otherwise going about our busi-ness, so what is actually essential? Is it simply facility with the four operationsof arithmetic on small numbers? In this chapter we encounter sweepingvisions of what learning and doing mathematics can offer young people inschool: recognition and development of the learners’ natural powers tomake sense of the world, forms of mathematical thinking that enhanceparticipation in society, and mathematical themes that pervade and linkdifferent mathematical topics. These global aims are consonant with the aimsof education and the perspectives on learning displayed in earlier chapters.

Using powers

Whatever powers learners possess, they need to be invoked, honed andrefined as part of their mathematical education.

Habits of mind: Al Cuoco, Paul Goldenberg and June Mark

Al Cuoco, Paul Goldenberg and June Mark have enormous experience inworking with learners using computer software such as LOGO which offerslearners the opportunity to construct mathematical objects for themselves.Here they advocate the sorts of ‘habits of mind’ that they would like learnersto develop as a result of working on mathematics.

… Take [the word should in what follows] with a grain of salt. When wesay students should do this or things like that, we mean that it would bewonderful if they did those things or thought in those ways, and thathigh school curricula should strive to develop these habits. We alsorealize that most students do not have these habits now, and that not

everything we say they should be able to do is appropriate for every situ-ation. We are looking to develop a repertoire of useful habits; the mostimportant of these is the understanding of when to use what.

(Cuoco, Goldenberg and Mark, 1996, p. 378)

There are similarities with knowing-to (see p. 289).

Students should be pattern sniffers: … In the context of mathematics, weshould foster within students a delight in finding hidden patterns in, forexample, a table of the squares of the integers between 1 and 100.Students should always be on the lookout for short-cuts that arise frompatterns in calculations … . … the search for regularity should extend totheir daily lives and should also drive the kinds of problems studentspose for themselves … .

Students should be experimenters: Performing experiments is centralin mathematical research, but experimenting is all too rare in mathe-matics classrooms. Simple ideas like recording results, keeping all butone variable fixed, trying very small or very large numbers, and varyingparameters in regular ways are missing from the backgrounds of manyhigh school students. When faced with a mathematical problem, astudent should immediately start playing with it, using strategies thathave proved successful in the past. Students should also be used toperforming thought experiments … [and] develop a healthy skepticismfor experimental results.

Students should be describers: Give precise descriptions of the steps ina process. Describing what you do it is an important step in under-standing it. … One way for students to see the utility and elegance oftraditional mathematical formalism is for them to struggle with theproblem of describing phenomena for which ordinary language descrip-tions are much too cumbersome … . … Students should be able toconvince their classmates that a particular result is true or plausible bygiving precise descriptions of good evidence or (even better) byshowing generic calculations that actually constitute proofs. … Studentsshould develop the habit of writing down their thoughts, results, conjec-tures, arguments, proofs, questions, and opinions about the mathematicsthey do, and they should be accustomed to polishing up these notesevery now and then for presentation to others.

(ibid., pp. 378–9)

There are similarities with Do–Talk–Record (see p. 262).

Students should be tinkerers: … Students should develop the habit oftaking ideas apart and putting them back together. When they do this,they should want to see what happens if something is left out or if thepieces are put back in a different way.


Students should be inventors: Tinkering with existing machines leadsto expertise at building new ones. Students should develop the habit ofinventing mathematics both for utilitarian purposes and for fun. Theirinventions might be rules for a game, algorithms for doing things, expla-nations of how things work, or even axioms for a mathematicalstructure.

Students should be visualizers: … doing things in one’s head that, inthe right situation, could be done with one’s eyes. … constructing visualanalogues to ideas or processes that are first encountered in nonvisualrealms. … visual accompaniments (not analogues, exactly) to totallynon-visual processes.

Students should be conjecturers: The habit of making plausible conjec-tures takes time to develop, but it is central to the doing of mathematics.

Students should be guessers: Guessing is a wonderful researchstrategy. Starting at a possible solution to a problem and working back-ward (or simply checking your guesses) often helps you find a closerapproximation to the desired result … [and] find new insight, strategies,and approaches.

(ibid., pp. 379–84)

Powers: Caleb Gattegno

Gattegno’s insights and suggestions were based on extensive observationand contemplation of the very young child’s experiences, at an age whenlearning is at its most extensive and fastest.

… I know that children can do a great deal more with themselves thanthe most adventurous educator ever dreamed of.

Children spontaneously stay with problems. And they stay for as longas is required. They consider abstraction (the simultaneous use ofstressing and ignoring) naturally as their birthright. They give proof thatthey know many concepts but, more than that, that they know how togenerate them in their awareness and how to recognise them as a repre-sentable by a word and as represented by an open class of elements towhich new items can be added. Their mastery of the language of theirenvironment in their tender years, whatever their language, tells usclearly that they can perceive mental structures as present in the mind,how the structures link to each other and how they affect each other.

[ … ]Children spontaneously transform. They spontaneously know that

everything is in flux and that a correct description of the world would bein dynamic terms. They know intimately that nothing is ever seen againin the same light, from the same distance and the same angle, and theobjects are classes of impressions which are defined with respect to eachother and by their overlapping parts. They know spontaneously that

Learning what? 183

conservation is not needed to make sense of a world in flux and system-atically ignore it until there is a place for it in their perception of theworld; for example, when quantity becomes an attribute of some quali-ties perceived within the field.

… The real structure behind equality and identity is … equivalence.Equivalence is the proper way of looking at the world because things aremade to belong to each other (in a class of equivalence) when we stressone or more attributes and ignore others. It is the stressing that unifies,but since the ignoring is also present, it serves to reopen a questionclosed by the stressing.


The teacher’s challenge: John Mason

Children enter school already having displayed immense powers ofimagining and expressing [describing what they see or imagine usinglanguage, displaying using their bodies, depicting], generalising andspecialising (in picking up and using language), and conjecturing andreasoning (detecting patterns in language so as to be able to make uptheir own sentences to express themselves). Exercising, developing andextending one’s powers is a source of pleasure and self-confidence.Failure to use those powers is at best throwing away an opportunity, andat worst, turns students off mathematics and off school.

So as a teacher I am faced with the question, ‘Am I stimulating mystudents to use their powers, or am I trying to do the work for them?’.

(Mason, 2002a, p. 107)

Mathematical thinking

It is one thing to assert that what we really want is for learners to developtheir mathematical thinking; it is quite another to be precise about just whatthat means. We may feel certain that learners who actively think about theirmathematics, who develop mathematical ‘habits of mind’, who use theirnatural powers, will also do better on examinations and tests than those whoare trained in a few techniques and typical problem types. There is evensome evidence in favour of this view (Boaler, 1997; Watson, 2001; Houssart,2001). But without some clarity about what we mean by mathematicalthinking, aims expressed in terms of it are likely to prove as empty as mostother general aims.


Becoming aware of your own powers and how you use them is an essentialstep towards planning lessons so that learners are stimulated to use their ownpowers.

Mathematics is … : Charles Saunders Peirce

Charles Saunders Peirce (b. Maryland, 1839–1914) was an influential Amer-ican scientist, philosopher and logician. He formulated the school of prag-matism which he called pragmaticism, to distinguish it from the pragmatismof William James (1842–1910) and Dewey.

It is difficult to decide between the two definitions of mathematics; theone by its method, that of drawing necessary conclusions; the other byits aim and subject matter, as the study of hypothetical states of things.

(Peirce, 1902, p. 1779)

In fact it was his father, Benjamin Peirce, who in 1870 had first defined math-ematics as ‘the science of drawing necessary conclusions’. ‘Another charac-teristic of mathematical thought is that it can have no success where it cannotgeneralize’ (ibid., p. 1778).

Later mathematician-philosophers such as Imre Lakatos (1976) and PhilipKitcher (1983) proposed that mathematicians are often empirical, in thesense of detecting patterns on the basis of examples, and refining andaltering conjectures and articulations of theorems many times before theysettle down.

Mathematics is … : Caleb Gattegno

Man’s mind is the generator of mathematics. Mathematics is a mentalactivity. Mathematical structures are mental structures.


This is quite a strong position, identifying doing mathematics with beingaware of mental functioning. In another book, Gattegno expanded on this:

[Mathematicians deal] with structures and relationships between struc-tures, rather than with so-called ‘mathematical objects’. In other words,mathematicians have become aware that the ‘objects’ of their science areparticular mental constructs to which they apply their mental dynamismin order to make explicit the content involved in them by virtue of theimbedded structures.

(Gattegno, 1963, pp. 39–40)

My first discovery was that mathematicians are specialists who givethemselves to becoming aware of relationships in themselves.

… The simplest illustration I know is the awareness I have of what I dowhen I climb steps. Everyone can recognise that the steps are of equalheight, that they succeed each other, that they can be climbed up or downfrom beginning to end and that these are variable. As soon as I extract

Learning what? 185

these awarenesses, make a note of each of them and invent a shorthandway of referring to them, I can produce condensed virtual expressions ofwhat I actually do in such activities. Such expressions looks strangely likethe formulae found in chapters devoted to arithmetic progressions.

(Gattegno, 1983, p. 34)

Mathematics emerges as the expression and formalisation of awareness ofawarenesses (such as climbing steps). The idea of making use of learner’sexperience in this way is also the core of Freudenthal’s didacical phenomen-ology approach (see p. 202).

Gattegno suggested how ‘mental objects’ are formed:

Everyone knows how habits are acquired by repeated drill and increas-ingly frequent check[s] to make certain that the aim set is achieved. It isthe interaction between check and drill that determines whether wecontinue the exercise or pass on to another activity, as can easily be seenfrom observation of children playing marbles, hopscotch, or leapfrog orthe pianist or the scientist at work. It is in the formation of mentalpatterns in which perception, action, and representation are intermin-gled that we find the mental structure. It is this that underlies the symboland is the basis of our social intercourse. It is this that is a starting pointfor elaboration into a more developed structure.

(Gattegno, 1963, p. 41)

There are similarities with how other authors describe how mental processesand actions become objects through reification (see p. 167).

Mathematical thinking as problem solving: George Polya

Polya was a brilliant mathematician who emigrated to the USA where hecontributed to the development of new mathematical ideas. Through histeacher education courses and his books on mathematical thinking, whichare full of wisdom and insight, he profoundly influenced mathematicsteaching in the USA. By reflecting on how he thought as a way of improvingthat thinking, he made reflection on and discussion about forms and types ofmathematical thinking respectable.

Getting food is usually no problem in modern life. If I get hungry at home,I grab something in the refrigerator, and I go to a coffee shop or someother shop if I am in town. It is a different matter, however, if the refriger-ator is empty or I happen to be in town without money; in such a case,getting food becomes a problem. In general, a desire may or may not leadto a problem. If the desire brings to my mind immediately, without anydifficulty, some obvious action that is likely to attain the desired object,there is no problem. If, however, no such action occurs to me, there is a


problem. Thus, to have a problem means: to search consciously for someaction appropriate to attain a clearly conceived, but not immediatelyattainable, aim. To solve the problem means to find such action.

A problem is a ‘great’ problem if it is very difficult, it is just a ‘little’problem if it is just a little difficult. Yet some degree of difficulty belongsto the very notion of a problem: where there is no difficulty, there is noproblem.

… the solution of any problem appears to us somehow as finding away: a way out of a difficulty, a way around an obstacle.

(Polya, 1962, p. 117)

Note the role of epistemological obstacles (see p. 303).Mathematical thinking is the use of natural powers of sense-making applied

to … ; to what? This is the difficulty, since it is hard to specify the content ofmathematics without ending up in a circular description: mathematicalthinking is applying powers to mathematical topics; mathematical topics ariseas the result of thinking mathematically! Polya plumps for problem solving,though it is just as hard to specify what makes a problem mathematical, orwhat it means to tackle a problem mathematically. The circle won’t go away!To break out of such a circle seems to require rather abstruse circumlocutions,much as Humberto Maturana, for example, who uses highly abstract languageto try to capture the essence of learning (see p. 70).

What is know-how in mathematics? The ability to solve problems – notmerely routine problems but problems requiring some degree of inde-pendence, judgment, originality, creativity. Therefore, the first and fore-most duty of the high school in teaching mathematics is to emphasizemethodical … problem solving.

(Polya, 1962, pp. xi–xii)

Mathematics as problem solving: Paul Halmos

… the mathematician’s main reason for existence is to solve problems,and that, therefore, what mathematics really consists of is problems andsolutions.

(Halmos, 1980, p. 519)

The major part of every meaningful life is the solution of problems; … Itis the duty of all teachers, and of teachers of mathematics in particular, toexpose their students to problems much more than to facts.

[ … ]One of the hardest parts of problem solving is to ask the right ques-

tion, and the only way to learn to do so is to practise.(ibid., pp. 523–4)

Learning what? 187

A teacher who is not always thinking about solving problems – ones hedoes not know the answer to – is psychologically simply not prepared toteach problem solving to his students.

(Halmos, 1985, p. 322)

Mathematical thinking: John Mason, Leone Burton and Kaye Stacey

Leone Burton (b. Australia, 1936–) and Kaye Stacey (b. Australia, 1948–) bothstudied and taught mathematics, but while Stacey maintains her roots in math-ematics, Burton has moved into more general educational issues and howthey pertain to mathematics education. In what has proved to be a classic text,Mason, Burton and Stacey put together advice on how to think mathematicallythrough engaging readers in the act of thinking for themselves.

Three kinds of involvement are required [to think mathematically]: phys-ical, emotional and intellectual.

Probably the single most important lesson to be learned is that beingstuck is an honourable state and is an essential part of improvingthinking. However, to get the most out of being stuck, it is not enough tothink for a few minutes and then read on. Take time to ponder the ques-tion, and continue reading only when you are convinced that you havetried all possible alleys. Time taken to ponder the question and to tryseveral approaches is time well spent.

(Mason, Burton and Stacey, 1982, pp. ix–x)

The authors went on to announce five assumptions: that everyone can thinkmathematically, and that mathematical thinking is provoked by contradiction,attention and surprise, supported by an atmosphere of questioning, chal-lenging and reflecting, and can be improved by practice with reflection.Finally, mathematical thinking helps in understanding yourself and theworld.

Thinking mathematically is not an end in itself. It is a process by whichwe increase our understanding of the world and extend our choices.Because it is a way of proceeding, it has widespread application, notonly to attacking problems which are mathematical or scientific, butmore generally. However, sustaining mathematical thinking requiresmore than just getting answers to questions, no matter how elegant thesolution or how difficult the question. …

(ibid., p. 154)

Awareness: Caleb Gattegno

One of Gattegno’s principal themes was the education of awareness. Inawareness he included things that the body does of which the mind is not


conscious, envisaging the person as a whole – body, mind and emotionstogether. He chose to describe subject matter in terms of awareness and howit might be educated.

As soon as we can shut our eyes and evoke images we can act virtuallyupon these images to make them do some of the things we want andwhich are compatible with some of the attributes retained in the images.

[ … ]… As soon as we see that because mathematics is of the mind we must

concern ourselves with the inner life of our students and, in particular …with their awareness of the dynamics of the mind, we cast an entirelynew light on the subject and on its transformation into life-giving activi-ties which contribute to each of our students’ personal evolution andhence a more responsible situating of themselves in the futuredescending upon us.

(Gattegno, 1983, p. 34–5)

Gattegno had a lot to say about awareness, because for him it was muchmore than consciousness. It is the basis of all our functioning, conscious andunconscious. To become a teacher, one must become aware of relevantawarenesses in oneself in order to be able to prompt learners to educatetheir own awareness.

[Teachers need to] make themselves vulnerable to the awareness ofawareness, and to mathematization, rather than to the historical contentof mathematics. They need to give themselves an opportunity to experi-ence their own creativity, and when they are in contact with it, to turn totheir students to give them the opportunity as well.

(Gattegno, 1988, p. 167)

Horizontal and vertical mathematisation: Hans Freudenthal and AdrianTreffers

It is not always easy to distinguish between pure and applied mathematics,since pure mathematics is the result of applying mathematical thinkingwithin mathematics itself. The distinction between concepts and applicationshas dogged mathematics education for centuries, despite various attempts toreunify them within school (for example, as real problem solving orauthentic mathematics, see p. 108). Theory first and application later hasbeen the practice for centuries (see p. 203). Treffers, a student and colleagueof Freudenthal, responded to this by making a distinction between hori-zontal and vertical mathematising:

Horizontal mathematisation leads from the world of life to the world ofsymbols. In the world of life one lives, acts (and suffers); in the other

Learning what? 189

one, symbols are shaped, reshaped, and manipulated, mechanically,comprehendingly, reflectingly; this is vertical mathematisation.


Freudenthal reported that initially he reacted against this distinction, but thatover time he came to see that it could be useful, as long as it was not taken asrigid and structural, but rather as partial and flexible. This lead him to extracttwo principles:

• Choosing learning situations within the learner’s current reality,appropriate for horizontal mathematising.

• Offering means and tools for vertical mathematising.(ibid., p. 56)

Mathematical modelling: Open University

In the 1970s, when modelling was being stressed as an important but over-looked aspect of learning mathematics, The Open University course M100introduced a seven-phase ‘model of modelling’ which was taken up in manyinstitutions:

The seven phases of modelling were identified in order to help learnersrecognise what they were doing, and to provide a structure for use if they gotlost. Notice the implicit correspondence between the three columns in theupper part of the figure and three worlds (see p. 73). Each of the phases canbe elaborated:

• Specifying the actual problem is not as simple as it sounds. Mathemati-cians like to be told the original problem and situation rather than tryingto work with someone else’s model.

• Setting up a model is about imagining the situation and stressing only someof the features and factors (keep it simple is a good motto), then locatingthe quantities and relationships which seem to be central to the problem.


• Formulating a mathematical problem involves moving to the world ofsymbols and equations.

• Finding a mathematical solution may involve altering the mathematicalproblem until it becomes solvable, or tractable in some way.

• Interpreting the solution involves interpreting alterations made in theoriginal, and identifying what the symbols represent.

• Comparing with the original situation involves checking to see if theproposed solution offers a satisfactory resolution of the original problemor whether the simplifications have rendered it too rough, in which casethe modelling cycle is repeated.

• When a satisfactory or good-enough resolution is reached, a report iswritten and the original problem resolved.

Although displayed as a cycle, there is often spontaneous to-ing and fro-ingbetween the various phases, or rather, between the mathematical world andthe world of the problem.

Mathematical modelling: John Mason

Mathematical modelling and, indeed, mathematical thinking, give structureto what we perceive, and also to how we perceive:

… the act of asking a question already comes from a way of perceiving,and so contains the seeds of its possible resolutions. For example it iswell known that when someone acquires a hammer, the world dividesitself into nails and non-nails: we perceive the world through theaffordances (Greeno et al., 1993) provided by the tools with which weare confident. …

Modelling is thus a co-emergent activity: the modeller perceivesproblematicity through the framework and structures of their past expe-rience, the tools which are extensions of their thinking, and theirdomains of confidence. A model emerges, sometimes extending thatexpertise, that facility, that perception, that sensitivity, those tools, andsometimes shaping what is perceived as problematic, as it develops.

Once formulated, a model is both frame and picture, both form andcontent. A particular model involves a way of seeing (the movementfrom one ‘world’ to another) as well as what is seen. Once constructed, amodel channels thinking through its structure, through its particularstressings and ignorings, as well as illuminating and supporting analysis.

(Mason, 2001, p. 39)

See also Wigner (p. 97).There are significant issues about presenting models to learners, engaging

learners in the act of modelling, and helping them to recognise it whenthey are doing it.

Learning what? 191

When a model is presented, there is more than just the model itself: thereare choices and assumptions to be made explicit and questioned, thereis the issue of whether the model adequately models the originalproblem, and there is the issue of whether the conclusions drawn areappropriate. Then there are the socio-political questions of who benefitsfrom this particular model being used. Students need to be introduced toall of these aspects of models. But we also present models in order tosupport students in appreciating how to go about modelling. So eachmodel acts as a case study for students. Or does it? …

[ … ]How does the student detect what is modelling, what is model, and

what is mathematics (in which world are they supposed to be oper-ating)? What is process and what is product? What is the studentsupposed to learn (memorise, learn from, learn about, … )?

(ibid., pp. 57–8)

There are similarities with the issue of when an example is exemplary forlearners (see Sierpinska, p. 173).

Modelling and mathematisation: David Wheeler

David Wheeler (b. Wiltshire, 1925–2000) was a teacher and mathematicseducator. He was, with Gattegno, one of the founder members of the Associ-ation of Teachers of Mathematics and he edited the journal of the associationfor many years. He then went to Canada and founded the journal For TheLearning Of Mathematics which is highly respected for its wide range ofthoughtful articles.

As already indicated, modelling is usually identified with setting up amathematical version of a situation in order to resolve some problem.Mathematisation is a more general process of perceiving the world throughthe lenses of mathematics: seeing relationships, properties and structures.Here Wheeler extols mathematisation.

It is more useful to know how to mathematize than to know a lot ofmathematics. Teachers, in particular, would benefit by looking at theirtask in terms of teaching their students to mathematize rather thanteaching them some mathematics.

(Wheeler, 1982, p. 45, quoting a 1974 paper of his)

Wheeler also observes that like speaking, mathematising is hard to catch as ithappens.

There are similarities between teaching mathematising and didactic phenom-enology (see p. 202) in the sense that the teacher constructs situations inwhich the learner can, perhaps with help and guidance, mathematise. Seealso using situations (p. 249).


Mathematical themes

There are a number of mathematical themes which pervade mathematicsand which have been identified by many different thinkers over the centu-ries. They provide connections between apparently disparate topics, firmingup a sense of connection arising because of and through the use of the samepowers and similar ways of working in apparently distinct topics.

Doing and undoing, reversing

If someone starts with a number which they do not reveal to you and thenperforms a sequence of calculations and tells you the answer, can you recon-struct the starting number by using their operations? If the operations can bereversed, that is, undone, then the number can be reconstructed, which isthe basis for solving equations. When you can get the answer to a question, itis useful to ask yourself, could you reconstruct the question. For example,3 × 4 = 12, but asking 12 = ? × ? leads to factoring and prime numbers. Findingthe remainder on dividing 15 by 7 is ‘doing’; finding all those numbers whichleave a remainder of 1 on dividing by 7 leads to algebra.

When a ‘doing’ leads to a particular, an ‘undoing’ is likely to lead to a collec-tion of particulars. Indeed, many of the insightful developments in mathematicshave arisen from people converting a doing-question into an undoing question.

Invariance amidst change

Most mathematical facts are actually statements about something being invariantwhile something else changes. The sum of the angles of a planar triangle isalways 180 degrees (invariant), no matter how the shape of the trianglechanges. The product of two odd numbers is always odd, no matter which oddnumbers are chosen. Whenever an assertion is encountered, it is worth askingyourself what it is that is permitted to change while leaving a particular invariantunchanged. This question is valuable because attention often is focused on theinvariant rather than on the permissible range of change of different features.

A single exercise or problem is a task. Asking yourself what could bechanged and still have the same approach or technique would apply opens updimensions-of-possible-variation (see p. 56); asking what sorts of change arepossible in any one such dimension opens up range-of-change, leading to thenotion of variable and parameter. Together these two ideas transform a set of

Learning what? 193

To think mathematically is to mathematise situations and apply (mathematical)powers in order to model situations inside and outside of mathematics itself. Itmeans to pose and resolve problems by following chains of necessary deductions,even if you sometimes work empirically in order to locate appropriate conjectures.

exercises from a sequence of tasks for learners to work through piecemeal, toa doorway into a domain of generality.

Invariance: Eugene Wigner

Eugene Wigner (b. Hungary, 1902–1995) emigrated to the USA in the 1930s.He was an applied mathematician who confessed to being amazed at howeffective mathematics is in modelling the material world (see p. 74 and p. 97).

… One … regularity, discovered by Galileo, is that two rocks, droppedat the same time from the same height, reach the ground at the sametime. The laws of nature are concerned with such regularities. Galileo’sregularity is a prototype of a large class of regularities. It is a surprisingregularity … .

The first reason that it is surprising is that it is true not only in Pisa, andin Galileo’s time, it is true everywhere on the Earth, was always true, andwill always be true. This property of the regularity is a recognizedinvariance property and … without [these] … , physics would not bepossible. The second surprising feature is that the regularity which weare discussing is independent of so many conditions which could havean effect on it. It is valid no matter whether it rains or not, whether theexperiment is carried out in a room or from the Leaning Tower, nomatter whether the person who drops the rocks is a man or a woman. Itis valid even if the two rocks are dropped, simultaneously and from thesame height, by two different people. There are, obviously, innumerableother conditions which are all immaterial from the point of view of thevalidity of Galileo’s regularity. The irrelevancy of so many circumstanceswhich could play a role in the phenomenon observed has also beencalled an invariance. However, this invariance … cannot be formulatedas a general principle. The exploration of the conditions which do, andwhich do not, influence a phenomenon is part of the early experimentalexploration of a field. It is the skill and ingenuity of the experimenterwhich show him phenomena which depend on a relatively narrow set ofrelatively easily realizable and reproducible conditions. …

(Wigner, 1960 webref)

There are similarities with dimensions-of-possible-variation (see p. 56).

Same and different: Colin Banwell, Dick Tahta and Ken Saunders

Banwell, Tahta and Saunders collaborated to produce a rich resource ofproblematic situations as the starting points for mathematical investigationand exploration, leading to most topics in the curriculum. It has served asinspiration for the many books that have followed, with most of the classictasks being used traceable back to this collection.


… mathematics has inevitably commenced when it is decided howthings are to be treated as the same. Among other issues, this raises thematter of how many different things there are. Equivalence is a choice atour disposal; but when a choice is made, counting must yield a uniqueanswer. Here is the essence of the activity. We are free to choose boundsand may then explore the inexorable implications of our choice. But theactivity must include the choice as well as the exploration.

(Banwell, Saunders and Tahta, 1972, p. 67)

Seeking ‘same and different’ is a potent way to explore invariance (see alsop. 127).

Freedom and constraint

Most mathematical problems begin with something (perhaps a number,perhaps a shape) which is indefinite or arbitrary. Constraints are thenimposed. With each imposed constraint, there is the question of whetherthere is sufficient freedom to permit some objects to meet that constraint.Thus, equations and inequalities are examples of constraints imposed uponinitial freedom.

Whenever learners are asked to find an answer, they are being asked toconstruct a mathematical object subject to some constraints. If instead of seeingmathematics as being about the getting of answers, it is regarded as a construc-tive enterprise, in which techniques are developed to facilitate some construc-tions, then perhaps more learners would appreciate the essential creativity inmathematics, and be able to experience the pleasure mathematics affords.

Ordering and classifying, stressing and ignoring

Putting things in order, which involves comparing and contrasting, can beseen in the play of very young children, in the organisation of collections ofadolescents, and in the structure of mathematics. We succeed as organismsbecause we can classify, that is, we can discriminate and recognise similari-ties and differences, which we do by stressing some features and ignoringothers. This ability to stress-and-ignore at the same time is the basis forspecialising and generalising, for imagining and expressing, and for orderingand classifying.

Extending and contracting meaning

When children first meet numbers, those numbers are whole, countingnumbers, and hence positive. Then they meet zero, and negatives, and frac-tions, and decimals. The word number expands with each new encounter.

When we do not know something, we tend to label it. So p labels a particularratio (circumference of any circle to its diameter: notice the invariance amidst

Learning what? 195

change). We use p to refer to the number even though we know just a fewbillion of its decimal digits. So we associate properties of a number with a name,even when the name is rather inexplicit. Similarly, 3 is a name for somethingwhich is positive and squares to 3: it is known only by its properties. Thus weextend the meaning of words and symbols as we learn new properties, or meetnew objects which also have the same properties (in the case of numbers).

We also focus meaning. Words used in common speech often have a tech-nical meaning in mathematics which is narrower and more precise. Thus ‘Heonly ate a fraction of the cake’ is not a mathematical use of fraction, becausethis idiom assumes a number between 0 and 1, whereas mathematically afraction can be the result of dividing any integer by any non-zero integer.

See also Davis and Hersh (1981).

Mathematical techniques and procedures

Teaching rules or revealing rules: Herbert Spencer

Spencer was an early advocate of using learners’ powers. Here he reports onthe demise of rote-teaching and the beginnings of getting learners to gener-alise for themselves.

Along with rote-teaching, is declining also the nearly-allied teaching byrules. The particulars first, and then the generalizations, is the newmethod … which, though ‘the reverse of the method usually followed,which consists in giving the pupil the rule first’ is yet proved by experi-ence to be the right one. Rule-teaching is now condemned as impartinga merely empirical knowledge – as producing an appearance of under-standing without the reality. To give the net product of inquiry withoutthe inquiry that leads to it, is found to be both enervating and inefficient.General truths to be of due and permanent use, must be earned. …While the rule-taught youth is at sea when beyond his rules, the youthinstructed in principles solves a new case as readily as an old one.

(Spencer, 1878, pp. 56–7)

Note the similarities with training behaviour and educating awareness(see p. 61 and p. 204). See also rote learning (p. 152) and instrumentalunderstanding (p. 295).


The study of mathematics may be helped by noticing the underlying themeswhich weave across topics. Recurring themes include Doing and Undoing,Invariance Amidst Change, Freedom and Constraint, Ordering and Classifying,Stressing and Ignoring, Extending and Contracting Meaning.

Inductive method: Warren Colburn

Warren Colburn (b. Massachusetts, 1793–1833) was a prolific author of arith-metic and algebra texts in the USA. In 1825 he published an algebra textbased upon what came to be called ‘the inductive method of instruction’,inspired by Herbert Spencer’s ideas. The term inductive came to refer tolearners detecting and expressing similarities and differences for themselves,and so reaching and expressing their own generalities. This, Colburn said,was in contrast to a deductive approach in which definitions, rules and prin-ciples were to be committed to memory, followed by a few illustrative exam-ples, and then applied to exercises.

The inductive approach began a movement to displace ‘recitation frommemory’ which had characterised much of mathematics learning for manycenturies (see p. 37). Instead, learners were exposed, using mental and thenslate-based exercises, to simple cases which built up to more complex cases.Following principles adumbrated by Pestalozzi (see p. 39), Colburn tried toengage the learner explicitly in making generalisations.

… The manner of solving the examples in each section is particularlyexplained. All the most difficult of the practical examples are solved insuch a manner, as to show the principles by which they were performed.Care has been taken to select examples for solution, that will explainthose which are not solved. Many remarks with regard to the manner ofillustrating the principles to pupils, are inserted in their proper places.

[…]The reasoning used in performing these small examples is precisely

the same as that used upon large ones. And when any one finds a diffi-culty in solving a question, he will remove it much sooner and muchmore effectively, by taking a very small example of the same kind, andobserving how he does it, than by [resorting] to a rule.

Colburn, 1863, pp. 109–10

Note the emphasis on observing yourself.See also practice makes perfect (p. 176) and discovery learning (p. 224).

For examples to be useful for induction they need to be seen as exemplary(see p. 173). Later texts merged the deductive and the inductive, suggestingthat a varied diet might be more effective. There may be no ‘best’ mixture ofmethods.

Learning what? 197

Rote learning is usually contrasted with understanding, but see p. 295. Teachersmight teach deductively or inductively: teaching rules first and applications later,or offering examples from which learners use their own powers with guidance.A mixed approach might be thought most likely to benefit a variety of learners.

Mathematical topics

Topics are specified by the curriculum, usually according to what will ulti-mately be examined. Under this heading we probe beneath the surface andlook for coherences within topics, and ways of thinking about topics whichcontribute to psychologising the subject matter (see p. 45 and p. 203).

First some questions:

• What is to be taught to whom? What is the nature of the differencebetween what they know and expect and what they want to know? Whatdifficulties and changes in meaning are likely to be required? What waysof working are familiar?

• When and under what condition (time of day, time of year etc.; energylevels of students, available technology, … ) is the teaching to takeplace?

• How (in what modes of interaction, in what sized groups, etc.)?• In what context (one-off; ongoing class or group; for examination prepa-

ration; for using powers; for sharpening techniques, … )?

Furthermore,

• Why might students choose to be taught? (Socio-institutional pressureand requirements; necessity to master tools, desire to understand,assessment driven; who benefits from learners being taught this topic?)

• What actions (shifts of attention, shifts of vocabulary, shifts of aware-ness) are required? What actions can be performed on what confidentlymanipulable objects?

• What activities would be likely to provoke requisite shifts? What specificgambits/devices might serve the purposes?

• What potentialities are present (for each task, each example, each act ofteaching)? What is the least and the most that are likely to happen; whatis fed by, and what feeds the functioning of each act? What is present inthe group?

• What modes of interaction are appropriate for the particular situation(students, content, teacher, environment, goals, topic and conceptimage)?

• What transformations are sought and how will they come about?

Concepts

Mathematical topics draw upon concepts which have proved fruitful forstating and resolving problems in the past. It is however very difficult to beprecise about just what a concept is!


Concepts: Hans Freudenthal

Freudenthal used an invitation to visit China to give a series of lectures as anopportunity to review a lifetime of investigation into the teaching andlearning of mathematics. Here he tackles the notion of concept, which soreadily slips into use in educational discussions.

Cognition does not start with concepts, but the other way around:concepts are the result of cognitive processes. Mathematics allowsexplicit definitions at an earlier stage than any other field of knowledge.For instance, ‘odd’ and ‘even’ can be defined on the basis of ‘wholenumber’. … But what about ‘whole number’? It is generated by aprocess, that of counting, rather than by an explicit definition, only tobecome a matter of common sense rather than a concept.

… it has become clear to increasingly more people that, where non-mathematicians are concerned, teaching the concept of X is not the appro-priate way to teach X. Cautious researchers now admit that concepts arepreceded by something less formal, by initiations, preconcepts, or whateverthey call it, which in the long run means that the proper goal is still that ofteaching concepts. In my view, the primordial and – in most cases for mostpeople – the final goal of teaching and learning is mental objects. I particu-larly like this term because it can be extrapolated to a term that describeshow these objects are handled, namely by mental operations.


Conceptual fields: Gerard Vergnaud

Gerard Vergnaud (b. France, 1933–) was a student of Piaget and is a leadingfigure internationally in research in psychology, with special attention to math-ematics. He is director of research of the National French Research Institute.

Piaget has demonstrated that knowledge and intelligence develop over along period of time, but he has done this by analyzing children’s develop-ment in terms of general capacities of intelligence, mainly logical, withoutpaying enough attention to specific contents of knowledge. It is the needto understand better the acquisition and development of specific knowl-edge and skills, in relation to situations and problems, that has led me tointroduce the framework of conceptual fields. A conceptual field is a set ofproblems … for the treatment of which concepts, procedures, and repre-sentations of different but narrowly interconnected types are necessary.

(Vergnaud, 1983, p. 127)

By delineating conceptual fields it is possible both to focus attention on arichly interconnected field of experience, and at the same time to becomeaware of significant differences between the thinking required in different

Learning what? 199

conceptual fields. Thus multiplication is not just repeated addition; indeed,seeing multiplication simply as repeated addition may prevent learners fromencountering the distinctive thinking required in the conceptual field ofmultiplicative structures. See Vergnaud (1982) for additive structures, andVergnaud (1983) for multiplicative structures.

Concept image: David Tall and Shlomo Vinner

David Tall (b. Northamptonshire, 1941–) is a mathematician-turned-educatorwho has led the study of advanced mathematical thinking as well as havingundertaken careful study of mathematics learning in secondary schools. ShlomoVinner (b. Jerusalem, 1936–) is also a mathematician-educator, and togetherthey have pursued the study of mathematics education at tertiary level.

What does it mean to ‘have a concept’? Tall and Vinner observed thatwhen a technical term is used, various images and propensities come tomind, giving access to ways of thinking and to specific techniques which amoment before had not been present. The totality of images and compe-tences which are associated with a term form the concept image of that term.

Many concepts which we use happily are not formally defined at all, welearn to recognise them by experience and usage in appropriatecontexts. Later these concepts may be refined in their meaning and inter-preted with increasing subtlety with or without the luxury of a precisedefinition. Usually in this process the concept is given a symbol or namewhich enables it to be communicated and aids in its mental manipula-tion. But the total cognitive structure which colours the meaning of theconcept is far greater than the evocation of a single symbol. It is morethan any mental picture, be it pictorial, symbolic or otherwise. Duringthe mental processes of recalling and manipulating a concept, manyassociated processes are brought into play, consciously and uncon-sciously affecting the meaning and usage.

We shall use the term concept image to describe the total cognitivestructure that is associated with the concept, which includes all themental pictures and associated properties and processes. It is built upover the years through experiences of all kinds, changing as the indi-vidual meets new stimuli and matures.

For instance the concept of subtraction is usually first met as a processinvolving positive whole numbers. At this stage children may observethat subtraction of a number always reduces the answer. For such a childthis observation is part of his concept image and may cause problemslater on should subtraction of negative numbers be encountered. Forthis reason all mental attributes associated with a concept, whether theybe conscious or unconscious, should be included in the concept image;they may contain the seeds of future conflict.

(Tall and Vinner, 1981)


Concept images can be the source of disturbances when incompatibleaspects are juxtaposed:

We shall call the portion of the concept image which is activated at aparticular time the evoked concept image. At different times, seeminglyconflicting images may be evoked. Only when conflicting aspects areevoked simultaneously need there be any actual sense of conflict orconfusion. Children doing mathematics often use different processesaccording to the context, making different errors depending on thespecific problem under consideration. For instance adding 1

2 + 14 may be

performed correctly but when confronted by 12 + 1

3 an erroneous methodmay be used. Such a child need see no conflict in the different methods,he simply utilises the method he considers appropriate on each occasion.

The definition of a concept (if it has one) is quite a different matter.We shall regard the concept definition to be a form of words used tospecify that concept. It may be learnt by an individual in a rote fashion ormore meaningfully learnt and related to a greater or lesser degree to theconcept as a whole. It may also be a personal reconstruction by thestudent of a definition. It is then the form of words that the student usesfor his own explanation of his (evoked) concept image. Whether theconcept definition is given to him or constructed by himself, he may varyit from time to time. In this way a personal concept definition can differfrom a formal concept definition, the latter being a concept definitionwhich is accepted by the mathematical community at large.

For each individual a concept definition generates its own conceptimage (which might, in a flight of fancy be called the concept definitionimage). This is, of course, part of the concept image. In some individualsit may be empty, or virtually non-existent. In others it may, or may not,be coherently related to other parts of the concept image.

(ibid.)

Image schemata: Willi Dörfler

Dörfler drew attention to an important constructivist awareness concerningthe role of drawings as carriers of meaning:

… an image schemata … cannot be shared with anybody else, only thecarriers can be communicated and in some cases the pertinent cognitivemanipulations have a corresponding material way of manipulating thecarrier. The image schema is just the specific way of viewing, interpreting,using, transforming, etc., the carrier. Therefore it is absolutely misleadingto regard the concrete carriers to ‘represent’ the respective concept. …much more appropriate to say that the individuals … present the conceptto themselves, make the concept present, cognitively and mentally.

(Dörfler, 1991, p. 21)

Learning what? 201

In other words, teaching is not a matter of constructing and presentinglearners with representations of ideas or ideals, but rather of bringing intotheir presence connections and images which then contribute to learners’construction of understanding (see also reification, p. 167).

Topic structure

Didactic phenomenology: Hans Freudenthal

Freudenthal’s central focus was the location and elaboration of phenomenawhich would, on the one hand, be familiar to learners at appropriate agesand, on the other hand, could lead to the exposure, elaboration, and formali-sation of the mathematical ideas commonly found in school mathematics. Heanalysed deeply the structure of individual mathematical topics from thepoint of view of the learner. He sought especially the prior experienceswhich the topic formalises, in order to construct tasks which would bring thetopic to the attention of learners, and from which they could, with guidanceand support, reconstruct the essential ideas themselves. His most famouswork captures this in its title, Didactical Phenomenology, relating as it doesto teaching based on learners’ prior experiences (Freudenthal, 1983).

Freudenthal first introduced what he calls inversion and conversion whichhe identified as ‘mathematical virtues’, as background to and justification forhis approach. Note the similarities with transposition didactique (see p. 83).

No mathematical idea has ever been published in the way it was discov-ered. Techniques have been developed and are used, if a problem hasbeen solved, to turn the solution procedure upside down, or if it is alarger complex of statements and theories, to turn definitions into prop-ositions, and propositions into definitions, the hot invention into icybeauty. This then if it has affected teaching matter, is the didacticalinversion … [which may not be helpful] … one should recognise that theyoung learner is entitled to recapitulate in a fashion the learning processof mankind. Not in the trivial abridged version, but equally we cannotrequire the new generation to start just at the point where their prede-cessors left off.

Our mathematical concepts, structures, ideas have been invented astools to organise the phenomena of the physical, social and mentalworld. Phenomenology of a mathematical concept, structure or ideameans describing it in its relation to the phenomena for which it wascreated, and to which it has been extended in the learning process ofmankind, … it is didactical phenomenology, a way to show the teacherthe places where the learner might step into the learning process ofmankind.

(Freudenthal, 1983, p. ix)


Note the similarities with Gattegno’s use of awareness (p. 61), and with theSoaT framework (see p. 203). Having distinguished between mental objectsand concepts (see p. 60), Freudenthal saw concepts as arising from themanipulation of mental objects. Freudenthal refers to Bruner’s three forms ofrepresentation (see p. 260). Then he criticises Dienes’ multiple embodiments(see p. 249).

The fact that manipulating mental objects precedes making conceptsexplicit seems to me more important than the division of representationsinto enactive, ikonic and symbolic. In each particular case one should tryto establish criteria that ought to be fulfilled if an object is to be consid-ered as mentally constituted. …

In opposition to concept attainment by concrete embodiments I haveplaced the constitution of mental objects based on phenomenology. Inthe first approach the concretisations have a transitory significance. Cakedividing may be forgotten as soon as the learner masters the fractionsalgorithmically. In contradistinction to this approach, the material thatserves to mentally constitute fractions has a lasting and definitive value.‘First concepts and applications afterwards’ as it happens in theapproach of concept attainment is a strategy that is virtually inverted inthe approach by constitution of mental objects.

(ibid., p. 33)

Freudenthal advocated starting with situations which are or can become‘real’ to learners through their experience (see realistic mathematics educa-tion, p. 110). He himself linked his notion of mental objects to Fischbein’sintuitions (see p. 63), saying that he prefered not to use Fischbein’s termbecause of other associations with it.

Structure-of-a-topic: The Open University

In seeking to provide a structure which would help teachers gain an over-view of a topic or concept, members of the Centre for Mathematics Educa-tion at The Open University devised a three-thread or three-axis structurebased on interweaving the traditional aspects of the psyche: enaction orbehaviour, affect and emotion, cognition and awareness. It arose throughreflection on their own concept images juxtaposed with the three aspects ofthe psyche (see p. 32) and informed by Gattegno’s memorable claim: ‘Onlyawareness is educable’.

Learning what? 203

The horizontal thread encompasses the motivational–emotional. Thisthread stresses both the contexts in which the idea or topic originallyarose, (even if it is a slightly fabricated or simplified story for students tounderstand), and in which it has been known to appear. It encompassesthe virtue of surprising students, whether by challenging a preconcep-tion, or indicating an unexpected result. One of the roles of this overallframework is to support locating the surprise which helped to turn thistopic from an ordinary result of perhaps passing interest into an actualtopic. …

The behavioural thread stresses the terms which students may alreadyknow or use in a less formal manner, language patterns (terms, phrases,clauses), both those which the students already use but less precisely, andthose which are the marks of competence and understanding. It includesalso specific manipulative techniques and any ‘inner incantations’ which arelative expert may employ when employing those techniques.

The awareness thread includes mental images, associations andconnections which the teacher would like the students to develop, tran-sitions from process to object which are entailed by or employed in thetopic or idea, as well as standard confusions or misconceptions whichstudents are likely to form because others have done so in the past. Thismay include historical analysis of obstacles encountered during theprecising of the topic or idea.

(Bachelard, 1938)

The image of interweaving the threads is intended to emphasise the impor-tance of mutual interaction and support between all three aspects. Theframework can be summarised by the expressions ‘harnessing emotion’,‘training behaviour’, and ‘educating awareness’, drawing on the centralimage of the psyche as analogous to a chariot (see p. 33). When one or moreof awareness, behaviour, and emotion is pushed to the background, studentexperience is impoverished, and learning made more difficult.

This framework is based on the image of a chariot and brings together many


of the ideas presented in previous chapters concerning three worlds (seep. 73), disturbance and surprise (see p. 55, p. 101 and p. 161), concept image(see p. 200), three aspects of the psyche (see p. 32), motivation (see p. 99),obstacles (see p. 303) and awareness (see p. 61 and 161), among others.

The Structure-of-a-topic framework was originally described in a series ofbooklets under the title of Preparing To Teach a Topic (Griffin and Gates,1989; see also Mason, 2002).

Developing thinking: Floyd et al.

In one of the courses prepared by the Centre for Mathematics Education atthe Open University in the 1980s, attention was drawn to the many differentlanguage patterns (and associated perspectives) which accompany a singletopic and to the ensuing abstraction. The next extract starts with thelanguage patterns for subtraction.

Surface understanding: difference between; more than, less than; takeaway; counting on/back; it’s the reverse of addition.

Deep understanding: subtraction is the root, common to all.Within each aspect of subtraction in turn, when children begin to

identify the one thing that all the activities have in common, and in spiteof variations in equipment or the game being played, they are beginningto abstract the particular subtraction aspect under consideration. Whenthey do the same thing across all the activities used in developing eachand every aspect of subtraction, they are distilling the very essence ofthe subtraction process itself. Only then can they be said to have devel-oped a sense of subtraction.

Implicit in all the foregoing is the view that, by structuring children’slearning experiences and by the skilful use of open-ended, self-orga-nizing questions, they can be guided to the point where they can and dodistil the essence of subtraction for themselves. If they don’t ask the nextquestion of themselves, they can be confronted with it: What do all theseactivities have in common?

(Floyd et al., 1982, pp. 15–16)

The children might need further prompting, but it is important that the childstill does the thinking.

The technique of juxtaposing a lot of specific cases like this makes iteasier for the learner to pull out the sameness. This is what [is meant] by‘consciously teaching’ a sense of subtraction. The fact that in all but onecase we started with the same numbers, 7 and 2, and produced the sameanswer, 5, indicates that the same process may be going on each time. Itis that process that we call subtraction. The one odd example is

Learning what? 205

explained by the insight that it is possible to find the answer to subtrac-tion sum by performing a related addition.

Generalizing from a lot of special cases is a central part of thinkingmathematically. By confronting children with these open questions andnot telling them what the sameness is you let them do the abstracting. Ifyou do this many times as part of a basic approach to teaching mathe-matical concepts, you’ll find the children begin to think in this way forthemselves. That is, they begin to develop mathematical thinking.

(ibid., p. 16)

There are similarities with same and different (see p. 128 and p. 194).

What learners learn

Learners are given tasks so that they will encounter and make sense of topics(including typical problems and techniques for solving them). So what dolearners actually learn from the tasks they are set? Not always quite what theteacher expects!

Learners do not always learn what is taught!: Margaret Brown andBrenda Denvir

Denvir and Brown studied changes in what low-attaining learners seemed tolearn as the result of being taught a particular topic. They concluded that, inthe short term, many learners actually seem to improve performance on avariety of tasks, not always those connected with what was taught. Over thelonger term, performance improved not only in connection with what wastaught, but also on other apparently unrelated items. This suggests thatlearning is a complex process which may not be enhanced by narrowteaching-and-testing, and certainly not usefully judged in this way.

1 Children must be active, both in interacting with the physical worldand in reflecting on these interactions.

2 Ideas and materials presented must be related to what childrenalready know, both to the types of reasoning available … and to their


The language of concepts may distance us from more important mentalobjects; concepts cluster around conceptual-fields; words denoting conceptsevoke concept images.

The Structure-of-a-topic framework incorporates and elaborates the notionof concept image and provides a structure to inform the preparation of lessonsconcerning any topic. The framework provides a collection of questions to useto interrogate the essence of a topic, to assist in psychologising the subjectmatter as Dewey would put it.

previous experiences … in order to achieve ‘meaningful learning’(Ausubel, 1978) and ‘relational understanding’ (Skemp, 1976).

3 In order to achieve integration of skills whereby new skills are devel-oped, there will need to be repetition of the mental process involvedin appropriate tasks.

4 In order to acquire mathematical concepts, children will need it, asDienes (1960) suggested, a variety of examples of those concepts indifferent mathematical forms, different contexts and, possibly, in thedifferent modes.

5 It is likely that as Bryant (1982) and Lawler (1981) suggest, childrenlearn when different intellectual strategies turn out to produce thesame result, especially if, in Lawler’s words ‘none was anticipated’.

(Denvir and Brown, 1986a, p. 144)

The study involved an analysis of mathematical and conceptual linksbetween ideas, displayed as a hierarchical network.

… the amount of progress made may be related to differences inresponse during the teaching programme.

1 This work supports the view that, in order to learn, the child needs toengage with the ideas in a manner and at a level which is meaningful.Such commitment can be encouraged but cannot be imposed by theteacher.

2 It appeared that whilst abler children may perceive relationshipswhich are not made explicit, the low attainer may need to engage inboth practical activities and discussion which explicitly draw atten-tion to such relationships.

3 The hierarchical framework can describe children’s present know-ledge and suggest which further skills they are most likely to acquireand thereby inform the design of teaching activities. However itcannot predict which skills or how many skills each child willacquire, so the teaching should not be too prescriptive or rigid in itsassumptions about what may be learned.

4 Children were not relaxed in the one-to-one teaching sessions.Whilst their attention needs to be focused on the mathematics, theirthoughts about it need to be spontaneous.

… [as a result] there were two major changes:

1 In order to promote discussion and to limit children’s self-consciousness, [the researchers] decided that pupils would be taughtin a group instead of individually. This would also have the advan-tage of being more readily transferred to normal classroom practice.

2 Children would be given general number activities which embodiedconcepts which they had already grasped as well as concepts which

Learning what? 207

they might acquire. Where appropriate, new skills or concepts wouldbe suggested but there would be a less specific intention to teachparticular points.

(ibid., pp. 156–7)

… the children did not always learn precisely what they were taught soattempts to match exactly the task to the child may not always have theexpected outcome. The effectiveness of group teaching depended onactivities being at a suitable level for all the children.

(ibid., p. 163)

Learners pick up teacher attitudes towards mathematics, and either assimi-late them, or react against them. They also pick up messages about what isrequired to minimise hassle and get through lessons. Some learners are moti-vated to work out what they have to do to succeed or to have an ‘easy life’.(Compare this with the hidden curriculum, p. 104.)

Misunderstandings

Learners inevitably make mistakes. They get hold of one aspect of an ideaand assume they have got it all or sometimes they get the wrong end of thestick entirely. But teachers can learn from learners’ mistakes, just as learnerscan (see also Balacheff, p. 252).

Mistakes: Alain Bouvier

Bouvier investigated the question of what to do about learners’ mistakes.

Students’ mistakes bother us. This assertion has become a sort of leit-motif at many meetings of mathematics teachers. Why is this? Do we seein students’ mistakes the sign of our failure?

The same mistakes appear very frequently, persisting across classes,resisting ‘correction’ week after week, as if there were certain‘invariants’, as if our teaching where unwittingly teaching mistakes. Whatis going on here? Why do some kinds of learning, like learning ourmother tongue, seem to be successful on a grand scale, while thelearning of mathematics is much less successful in spite of many hours ofteaching spread over many years?

Let us take a simple example, the calculation of 23. Among the possibleresponses, two are common: 23 = 6 and 23 = 5. In many cases we have onlyto ask a student who has written one of these, ‘Are you sure?’ or ‘Why?’,for him to change his response immediately to the expected answer.

(Bouvier, 1987, p. 17)

There are similarities with reversal (see p. 178).


Some teachers may say that the student made a careless mistake; but,especially since Freud, we know that the smallest ‘slip of the tongue’ hassignificance. Moreover, although we may meet 23 = 6 or 23 = 5, we neverseem to meet 23 = 37 or 23 = 100 [or 23 = 2000?]. Mistakes are not the resultof chance. They show that the student has used a particular logic,though not the appropriate one.

Others may say that the student ‘confused’ 23 with 2 × 3 = 6. Suchexplanations explain nothing! Do you know any child who would …confuse two drawings [a house and a boat] … ?

(ibid., p. 17)

Bouvier went on to suggest that there was a need to study learners’ concep-tions (concept images, see p. 200), and that to use these effectively theteachers need to study their own concept images.

Mistakes: Caleb Gattegno

Gattegno was adamant that learners’ mistakes are the responsibility of theteacher, and a vital source of stimulation and information for teachers.

Though obviously we ultimately come to the point at which the mistakesmust be corrected and the possibility of their recurrence eliminated, Iwould suggest that we should do well to curb our tendency to correct,and develop the habit of incorporating into our lessons the observationsthat we cannot fail to make in marking homework or in using an oralapproach with our classes.

It is man’s privilege to make mistakes; only through experience, expe-rience that is often painful, does man learn and acquire some degree ofwisdom. In the teaching of mathematics, the opportunity for gaining trueunderstanding through experience is too often reduced to the minimum.There is always someone who knows, who can produce the rightanswer, which is imposed upon those who cannot. But how often mustthe teacher make the same correction, and how many children reach theend of their school career under the impression that only their morefortunate fellows who are ‘mathematical’ can hope to avoid the mistakeswhich to them are inevitable?

(Gattegno, 1963, pp. 23–4)

Note parallels with learning from experience (see p. 263). The Structure-of-a-topic framework (see p. 203) follows Gattegno in suggesting collecting learnermistakes and conceptions with each topic in preparation for future planning.

As an example, the comprehensive study undertaken in the UK in the1980s, under the heading of CSMS (Concepts in Secondary Mathematics andScience, Hart et al., 1981) started with a test administered to hundreds ofsecondary students. Their errors were often so interesting that follow-up

Learning what? 209

interviews were conducted, revealing all sorts of clever conjectures thatlearners make, and curious conclusions that they come to in their attempts tomake sense of what they (think they) heard or saw in lessons.

Bugs: Kurt van Lehn

Van Lehn, with his supervisor, John Seeley Brown, adopted a computer-based perspective when investigating children’s errors in subtraction.Human beings do not usually ‘crash’ when a problem arises, rather theywork around it. Van Lehn and Seeley Brown explored the possibility thatlearners pieced together fragments of procedures picked up at differenttimes and in different situations. By classifying such sub-procedural frag-ments they were able to account for about 60 per cent of children’s perfor-mances on subtraction problems. The key to their approach is that children’serrors are far from wilful. Rather the errors are due to attempts to use frag-ments of procedures partly remembered.

The procedure-following assumption: During test-taking situations,students solve arithmetic problems by following a procedure (or tryingto) rather than by searching for a sequence of operators to transform theproblem’s initial state into a final, desired state.

To put it another way, the knowledge that students carry into the test-taking situation (or any other situation where an expert is not readilyavailable) consists of a procedure (or plan or search control knowledge)and perhaps some weak specifications of the syntax of the final state (forexample, how long the answer should be). When their procedure fails touniquely specify what to do next (that is, they reach an impasse), then itwould be irrational of them to abandon their procedure and rely exclu-sively on their final state description because, in the case of subtraction,that final state description is very weak. Thus even if search controlknowledge fails, then students probably do not search in the traditionalsense of finding operations to achieve a known final state. Rather theyperform a search of a different kind, which seeks an expedient way toovercome the impasse and return control to the procedure/searchcontrol knowledge. Thus the motive behind the students’ activity is notso much to achieve a final state, because they do not have a good speci-fication of what that is. Rather their motive is to follow a procedure asclosely as possible.

(van Lehn, 1990, pp. 36–7)

Students do not have knowledge of the design … of mathematicalcalculation procedures. Their knowledge consists only of the procedureitself.

(ibid., p. 40)


Van Lehn suggested four kinds of impasse:

• Decision impasses: the procedure calls for a decision which cannotcurrently be made. The procedure has to be suspended in order toresolve the impasse

• Reference impasses: lack of unique identity of an object being referred to.• Primitive impasses: descriptions of procedures always have primitive

acts. They are mentioned but not described by the procedure. It isassumed that one already knows how to carry out pursuant primitiveactions. If a production cannot be performed, then the impasse is classi-fied as a primitive failure.

• Critic impasses: a critic is triggered when the procedure is about to dosomething that is known to be wrong. For instance, one possible criticfor subtraction is that all the digits in the subtrahend and the minuendmust be used at some time during the course of problem solving. If theprocedure tries to finish before using all the digits, the critic triggers, andan impasse occurs. (Based on van Lehn, 1990, p. 41.)

What happens when an impasse is encountered?

The impasse-repair assumption: when people reach an impasse whileexecuting a procedure, they treat the impasse as a problem, solve it, andcontinue executing the procedure. The solution to an impasse is called arepair. …

(van Lehn, 1990, p. 42)

The common-knowledge assumption: repair strategies are task-generalmethods, most of which are familiar to most subjects.

(ibid., p. 43)

The impasse-repair independence assumption: subject strategies forselecting repair strategies at impasses are so variable that aggregateselection data can be approximated by random choices that are inde-pendent of the type of impasse and the surrounding situation.

(ibid., p. 53)

The patches assumption: subjects knowledge of a procedure mayinclude patches [repairs].

(ibid., p. 58)

These assumptions were the subject of van Lehn’s inquiry. He and Brownwere able to account for some 60 per cent of learner subtraction errors interms of compound procedures assembled from fragments of taughtprocedures.

Learning what? 211

Learning from mistakes and misconceptions: Pearla Nesher

Pearla Nesher (b. Israel, 1930–) is a psychologist with particular interests inlearning. She is a leading researcher in mathematics education in Israel. Hereshe considers the claim ‘we learn from our mistakes’.

We hold many beliefs that we are unaware of and which are part of ourhabits, yet once such a belief clashes with some counter-evidence orcontradictory arguments, it becomes the focus of our attention andinquiry.

[ … ]I found it very refreshing when visiting a second grade class to hear

the following unusual dialogue:Ronit: (second grader with tears in her eyes): ‘I did it wrong’ (referring toher geometrical drawings). ‘Never mind’, said the teacher, ‘what did wesay about making mistakes?’. Ronit (without hesitation) answered: ‘Welearn from our mistakes’. ‘So’, added the teacher, ‘don’t cry and don’t besad, because we learn from our mistakes’.

The phrase ‘we learn from mistakes’ was repeated over and over. Theatmosphere in the classroom was pleasant and the use of this phrase wasthe way the children admitted making errors on a given task. At thispoint I became curious and anxious to know what children really didlearn from their mistakes. … The exercises consisted of a given shapeand given axis of reflection … ; the children first had to hypothesize (orguess) and draw the reflected figure in the place where they thought itwould fall, and then to fold the paper along the reflection axis and bypuncturing the original figure with a pin to see whether their drawingwas right or wrong.

[ … ]… what did the children really learn from their mistakes? When each

child who made an error was asked to explain to me what was learnedfrom his or her mistake I could not elicit a clear answer. Instead the chil-dren repeated again and again that one learns from mistakes in a waythat started to sound suspiciously like a parroting of the teacher’s phrase.At this point it became clear to me that the teacher tolerated errors, butdid not use them as a feedback mechanism for real learning on the basisof actual performance.

(Nesher, 1987, p. 34)

Misconceptions can become a source for learner errors:

… the notion of misconception denotes a line of thinking that causes aseries of errors all resulting from an incorrect underlying premise, ratherthan sporadic, unconnected and non-systematic errors. It is not alwayseasy to follow the child’s line of thinking and reveal how systematic and


consistent it is. Most studies, therefore, report on classification of errors[and difficulty], though this does not explain their source and thereforecannot be treated systematically. Or, when dealt with, it is on the basis ofa mere surface-structure analysis of errors … . When an erroneous prin-ciple is detected at [a] deeper level it can explain not a single, but awhole cluster, of errors. We tend to call such an erroneous guiding rule amisconception.

[ … ]… misconceptions are hard to detect. This is because on some occa-

sions the mistaken rule is disguised by a ‘correct’ answer. That is, thestudent may get the ‘right’ answer for the wrong reasons … .

(ibid., pp. 35–6)

The language of misconceptions has never settled comfortably beside a viewof learners as making the best sense they can of their experience. Misconcep-tions can only be so labelled by someone who has different, perhaps moresocially accepted, conceptions or conceptions more consistent with otherconceptions. (See also diagnostic teaching, p. 233.)

Learning what? 213

Errors are part of learning, indeed, discerning conflict between expectationand experience or between conjecture and evidence is a principal stimulus tolearning (see disturbance, p. 55, and conjecturing, p. 139). Attending to theslips and conjectures made by learners reveals a good deal about their sense-making processes. Furthermore, common errors and misconceptions can serveas the basis for useful tasks (see conflict-discussions, p. 234).

Section 2

Guiding anddirecting learning

The previous section described learning as a process in which learners arethemselves active. What then is the role of the teacher? There are manydifferent teacher actions: proposing tasks; asking questions; telling learnersthings they need to know; initiating discussion; commenting on work; evalu-ating; urging learners to practice in order to automate techniques. All ofthese actions are, of course, dedicated to stimulating and prompting learnersto take initiative, and to become mathematically active. There is a multitudeof additional teacher actions involving discipline and institutional practicessuch as homework setting and counselling, as well as preparation and recordkeeping, which are beyond the scope of this book. The extracts chosenconcentrate on the choice and structure of mathematical tasks, and on whatpossibilities they afford for pedagogical interaction between teacher andlearner.

In proposing a mathematical task, teachers have a sense of what they wantor expect the pupils to do with it and get from it. They are aware of whataspects they intend to stress, and where it fits into the work of the day, term,year, as well as into any legislated description such as a national curriculum.What makes one teacher stress one aspect and a second stress another? Howdo pupils work out what is expected of them?

This section looks at the heart of teaching, specifically the subtle and notso subtle interactions with learners which contribute so much to their experi-ence and yet which are so hard to analyse or research effectively. Whatfactors influence which learners, and in what ways? As Freudenthal said, ‘If alearning process is to be observed, the moments that count are its disconti-nuities, the jumps in the learning process’ (Freudenthal, 1978, p. 78). Whatthen can the teacher do? The extracts provide a number of differentconstructs for addressing these questions.

8 Teachers’ rolesTeachers’ roles

Introduction

Teaching is a highly complex activity, a fact not always appreciated by thosewho look at it from a distance. The first problem a teacher has is to surmountthe negativity that surrounds teaching. Teachers and ‘current teachingmethods’ are always the focus of complaint when an educational crisissurfaces. As early as 1928, Philip Ballard observed that:

Rarely is a teacher satisfied with the arithmetic of his class. He is notallowed to be. Somebody always has something to say about it – generallysomething unpleasant. If it is not a colleague, it is probably an inspector; ifit is neither of these, it is a parent or an employer. … What the teacher hasto do is to grow a skin thin enough to let him know when he is hit, butthick enough to protect him from serious wounds. …

(Ballard, 1928, p. ix)

Positioning

Complexity of teaching: Magdalene Lampert

Lampert studied her own teaching in fine detail over a full year with the aimof exposing the highly problematic nature of teaching.

By taking a close look at the actions of a single teacher, teaching a singlesubject to a whole class over an entire academic year, I attempt to iden-tify the problems that must be addressed in the work of teaching.Considering the nature of a teacher’s actions as she addresses theseproblems, I try to explain what it is that is so hard about the work ofteaching and what we might mean when we call this practice ‘complex’.The single teacher I study here is myself. …

Teachers face some students who do not want to learn what they wantto teach, some who already know it, or think they do, and some who arepoorly prepared to study what is taught. They must figure out how to

teach each student, while working with a class of students who are alldifferent from one another. They have a limited amount of time to teachwhat needs to be taught, and they are interrupted often.

(Lampert, 2001, p. 1)

One reason why teachers are so often criticised is that they form the interfacebetween society and young people; they are the agents through whichsociety exercises control and direction over the coming generation. Conse-quently, society’s ills are reflected sharply in the classroom. However, class-rooms are thought to be where young people are most influenced; althoughthat influence may be significant, it is often subtle and hard to predict.

Positioning: Rom Harré

Harré (b. New Zealand, 1927–) has had a varied career as a historian ofscience and a sociologist, as well as working in artificial intelligence. Hisbooks, no matter what the subject, are influential standards. As a sociologist,Harré with his colleague Luk van Langenhove (Harré and Langenhove, 1992;1999) introduced and promoted the language of positioning to replace thelanguage of roles: people are positioned by what other people say and do.You can be positioned as the ‘one who knows’, or as the ‘one who neverknows’, as the ‘silent one’ or as the ‘group clown’, sometimes wittingly, andsometimes unwittingly, all as the result of group interaction. Sometimes posi-tioning is temporary, but sometimes it is robust against change. Usuallypeople do not realise how they are being positioned: their actions andthinking adjust to the roles and perspectives. Nor is it all one-way. Groups ofpeople position individuals in the group, but the positioning of self andothers evolves through the willing and unwilling responses of those individ-uals within the group.

Adolescents and children in institutions called schools immediately posi-tion the teacher as authority and warder: source and enemy. Learners arepositioned as novices in general, and especially as low or high achievers, as‘good at this but not at that’, as deficient in some ways and having strengthsin others, and so on. How parents and teachers perceive individual learnersand groups of learners exerts tremendous influence on those learners. Some-times they react by striking out, often in unexpected and even unconnectedways; sometimes they go along with the positioning.

Positioning: Derek Edwards and Neil Mercer

Edwards and Mercer (see p. 97) have studied the language used betweenlearners and between teachers and learners, trying to see how and underwhat circumstances learners and teachers position each other in class-room interactions.


It is largely within the teacher–pupil discourse through which the lessonis conducted that whatever understandings are eventually created are inthe first place shaped, interpreted, made salient or peripheral, reinter-preted, and so on. And it is a process that remains essentially dominatedby the teacher’s own aims and expectations … .

(Edwards and Mercer, 1987, p. 126)

Stances: Jerome Bruner

Bruner used the slightly different language of stances which learners andteachers might take or find themselves positioned into taking in variousdifferent classroom situations.

Each fact we encounter comes wrapped in stance marking . . . Somestances are invitations to the use of thought, reflection, elaboration,fantasy . . . if the teacher wishes to close down the process of wonderingby flat declarations of fixed factuality, he or she can do so. The teachercan also open wide a topic . . . to speculation and negotiation.

(Bruner, 1986, p.126)

Examples of stances taken by teachers are captured in the catchy phrases aguide on the side in contrast to a sage on the stage (King, 1993), a source ofresource, a being in question, and discovered not covered (Halmos, 1980, p.523). Any single stance used exclusively is likely to become overly familiarand boring, while too much chopping and changing may prove unsettlingfor some learners.

Bruner’s view, inspired by Vygotsky’s theories (see p. 84), developed in anumber of similar but different directions, under labels such as cognitiveapprenticeship, enculturation, situated cognition, authentic mathematics,and real problem solving. Other perspectives, drawn from different sources,include investigative teaching and inquiry methods. Each of these labels is aform of stance or perspective which involves a way of describing interactionsbetween teachers and learners, and which can be used to identify certainstyles of interaction. Roles and stances are labels applied by observers; posi-tioning is what happens in situations.

Teachers’ roles 219

Teachers operate with a range of positions or stances, in a complex socialsetting, which impacts significantly on learning.

Teachers, learners and mathematics

Teaching triads

Teachers teach learners mathematics, so there are three elements present inorder to have a teaching situation: teacher, learner, and mathematics. Since itall takes place within a milieu (see p. 95), the triad has to be seen as oper-ating within a context. Magdalene Lampert usefully elaborated on the triad ofteacher–student–content by analysing the forces acting on and within eachcomponent (Lampert, 2001).

For some people, the context is not only vital, but is dominant (see socialconstructivism, p. 95). But there are other ways of looking at teaching.

A teaching triad: James Stigler and James Hiebert

Stigler and Hiebert were leading researchers in the international TIMSS study(see p. 40). International comparisons based on tests given to learners ofroughly the same age led governments all over the world to institute curriculumreform, and led researchers to try to account for differences in performance.

In Japanese lessons, there is the mathematics on one hand, and thestudents on the other. The students engage with the mathematics, andthe teacher mediates the relationship between the two. In Germany,there is the mathematics as well, but the teacher owns the mathematicsand parcels it out to students as he sees fit, giving facts and explanationsat just the right time. In US lessons, there are the students and there is theteacher. I have trouble finding the mathematics; I just see interactionsbetween students and teachers.

(Stigler and Hiebert, 1999, pp. 25–6)

In some cases, teachers are in charge of the mathematics, and the mathe-matics is quite advanced, at least procedurally. Teachers often lead learnersthrough a development of procedures for solving general classes of prob-lems. Emphasis is on the technique, including both the rationale and theprecision with which the procedure is executed. In other cases, teachersappear to take a less active role, allowing learners to invent their own proce-dures for solving problems. These problems can be quite demanding, bothprocedurally and conceptually. Problems are chosen to use proceduresrecently developed. In other instances, teachers present definitions anddemonstrate procedures for solving specific problems; learners are thenexpected to memorise definitions and to practise procedures (based onStigler and Hiebert, 1999, pp. 26–7).

Many teachers have set out to try to capture the essence of ‘theirapproach’, and to outline what makes particular approaches likely to beeffective (see, for example, Mathematics Teaching, 139, 1992). There is,


however, no totally convincing research to show that any one approach isbetter than others, only that when teachers are actively working on theirteaching, encouraging learners to take an active part in lessons, the learnerstend to do better on tests as well.

Six modes of interaction: John Mason

Tasks are what make subsequent interaction between teacher and learnerpossible. Useful activity is the means whereby the learner encounters signifi-cant mathematical ideas and themes, as well as, perhaps, aspects of them-selves, as described by the inner aspects of a task (see p. 241). For activity tobe meaningful, some deeper action is necessary. The triad of teacher–learner–mathematical content, (or tutor–student–content) all within a milieuor environment (see p. 95) gives rise to an analysis of six different modes ofinteraction, based on ideas developed by J. G. Bennett (1956–1966; 1993).Three terms (teacher, learner, content) can occupy three roles (initiating,responding and mediating) which are needed for relationship and action tooccur:

In order to have what I call a balanced learning environment, there aresix activities … or modes of student–tutor[–content] interaction:

Expounding Exploring Exercising

Explaining Examining Expressing

The columns are characterised by the source of the initiative: tutor,student and content. Since this is an unusual way of describing learningactivities, a few words about each are in order.

The first two modes, ‘expounding’ and ‘explaining’ are on the initiativeof the tutor. The tutor expounds in the form of a lecture or in writing a textor in much of what we call tutorials. In a tutorial, a student asks a questionand very quickly the tutor lapses into a lecturing mode. This can easily bedetected in the tone of the tutor’s voice. In a true tutorial explaining takesplace when just one student is involved, with the tutor being perceptive ofthe student’s individual needs. This is rather more difficult than it sounds! Ihave often found myself trying to drag a student into my conceptual worldrather than trying to enter and remain in that of the student.

The next two modes depend on the initiative of the student: ‘explor-ing’ and ‘examining’. At some point initiative must pass to the studentwho begins to explore open-ended problems, to generalise for himself,and to begin what we call research. The role of the tutor is radicallydifferent from the previous modes because here his aim is to guide thestudent, to stimulate independent thought … . The idea of projects andguided investigations are attempts to make use of the exploring mode.

(Mason, 1979, p. 32)


Being aware of the different modes of interaction can assist in: staying with‘explaining’ in tutorial mode rather than sliding into expounding; looking foropportunities to amplify learner experience of the desire to express and toexercise; getting learners to use their powers in creating and exploring.

There are other approaches to exploring such as inviting learners to make uptheir own problems like those used in class, or inviting them to construct exam-ples of objects that meet certain constraints (see Watson and Mason, 2002).

It may be surprising to find ‘examining’ under student initiative, but thisis where I truly believe it belongs. … a candidate submits himself forexamination when he considers himself ready. … an opportunity for astudent to validate his own criteria of whether or not he is understandingthe material. After all, a major part of teaching is to provide the studentwith his own criteria. …

The final two modes require the initiative to come from neither thestudent nor the tutor, but rather from the content which connects thetutor and the student: ‘exercising’ and ‘expressing’.

Of course when we are looking at the situation we think we see theinitiative coming through the student. I say that the initiative for exer-cising really comes from content but expresses itself through the student… . It is experienced as a force inside me, yet all embracing, and is quitedifferent from a quick decision which doesn’t last. There are techniquesthat must be mastered and there are concepts that need practice inrecognising and manipulating. But it is defeating the purpose of theexercises to force the student, just as trying to push people to take phys-ical exercise does not work [unless] they have a strong inner force whichattracts them to it. Inappropriately forced, practice of academic materialusually results in the kind of rote learning which so annoys teachers. …

The final mode is widely recognised but little acted upon. We allknow that the best way to learn something is to try to express it toothers, yet it is difficult to get students to express their [current] under-standing. Indeed it is difficult enough to get them to ask a real question,one which has arisen inside them from significant contact with the mate-rial and which they have truly pondered.

(Mason, 1979, p. 32)


The triad of teacher–student–content operates within a milieu and includes sixdifferent modes of possible interaction.

Tasks and classroom interactions cannot fruitfully be isolated from thewhole situation in which they are embedded. Teachers with learners candevelop classroom ethos or atmosphere, which means ways of working onmathematics, and ways of working with each other.

Teaching as …

Guiding: John Holt

Holt discovered through close reflection on his own practice that his goalswere not always the same as those of the learners in his classes:

… I used to feel that I was guiding and helping my students on a journeythat they wanted to take but could not take without my help. I knew theway looked hard, but I assumed they could see the goal almost as clearlyas I and that they were almost as eager to reach it. … They were inschool because they had to be, …

(Holt, 1964, pp. 37–8)

Guiding and reinvention: Hans Freudenthal

Addressing the question of how children as learners can becomeenculturated into the richness of a (mathematical) culture which has devel-oped over thousands of years, Freudenthal suggested:

Children should repeat the learning process of mankind, not as it factu-ally took place but rather as it would have done if people in the past hadknown a bit more of what we know now.

New generations continue what their forbearers wrought but they donot step in at the same level reached by their elders. …

… guiding reinvention means striking a subtle balance between thefreedom of inventing and the force of guiding, between allowing thelearner to please himself and asking him to please the teacher. Moreoverthe learner’s own free choice is already restricted by the ‘re’ of‘reinvention’. The learner shall invent something that is new to him butwell-known to the guide.

(Freudenthal, 1991, p.48)

Note the similarity with the teaching dilemma (see p. 96).

… the learner should reinvent mathematising rather than mathematics;abstracting rather than abstractions; schematising rather than schemas; forma-lising rather than formulas; algorithmising rather than algorithms; verbalisingrather than language – let us stop here, now that it is obvious what is meant.

(ibid., p. 49)

Note the similarity with developing and extending learners’ natural powers(see p. 115).


If the learner is guided to reinvent all this, then valuable knowledge and abili-ties will more easily be learned, retained, and transferred than if imposed.

(ibid., p. 49)

Guiding means striking a delicate balance between the force of teachingand the freedom of learning. It depends on such a perplexing manifold ofhardly retrievable and only vaguely discernible variables that it seems inac-cessible to any general approach. Observational reports on guiding may bea source of understanding and a help for teaching guidance. Unfortunately,most of the reports available are concerned with simple lessons or shortsequences, and little is known about long-term learning processes.

(ibid., pp. 55–6)

Discovery learning: Government reports

‘Discovery learning’ grew out of the work of Dewey, Bruner and others. ‘Dis-covery learning’ has had multiple interpretations, ranging from ‘learners haveto discover everything for themselves with a minimum of guidance’, to ‘theteacher guides and cajoles the learners into rediscovering’. Consequently therehas been considerable criticism, as well as support for ‘discovery learning’:

[there is] unchallengeable evidence that sound and lasting learning can beachieved only through active participation. … Although the discoverymethod takes longer in the initial stages … far less practice is required toobtain and maintain the efficiency in computation when children havebeen able to make their own discoveries. … When children explore forthemselves they make discoveries which they want to communicate to theirteacher and to other children and this results in frequent discussion. It is thischange relationship which is the most important development of all.

(Schools Council, 1965)

Integrated teaching: Heinrich Bauersfeld

Bauersfeld contrasted discovery learning with integrated teaching:

Discovery approach: In explicitly defined situations, the student ‘researcher’starts off from an introduction to working on prepared material, andfinally ends up discussing and sharing findings in a whole-class session.

Integrated (culture) approach: In every classroom situation, the studentsare expected to search for patterns, to assume regularities, and torelate developing or contrasting ideas, as well as to give reasons andarguments for the issue under discussion.


See also scientific debate (p. 277) and conjecturing (p. 139).


Learning through problems: Magdalene Lampert

Lampert’s approach to teaching was to use problems. Her problems were notmere exercises, but problems arising from many different contexts. There arestrong similarities with investigative teaching which flourished in the UK inthe 1980s (Boaler, 1997; Ollerton and Watson, 2001), and various projects indifferent countries. Variations on this theme can be found in every genera-tion over a hundred years or more.

It is worth recalling Polya’s phases of learning (see p. 146):

Let us summarize: for efficient learning, an exploratory phase shouldprecede the phase of verbalization and concept formation and, eventu-ally, the material learned should be merged in, and contribute to, theintegral mental attitude of the learner.

This is the principle of consecutive phases.

(Polya, 1962, part 2, p. 104)

Listening: Brent Davis

Davis extolled the virtues of listening as a means of teaching:

Occurring somewhere between the surety of the known and the uncer-tainty of the not-yet-known, the act of listening is similar to the project ofeducation. It is, after all, when we are not certain that we are compelledto listen. Our listening is always and already in the transformative spaceof learning.

(Davis, 1996, p. xxiv)

He went on to identify and develop three different forms of listening:

Evaluative listening: … Within the mathematics classroom, this mannerof listening is manifested in the detached, evaluative stance of theteacher who deviates little from intended plans, in whose classroomstudent contributions are judged as either right or wrong (and thushave little impact on lesson trajectories), and for whom listening isprimarily the responsibility of the learner.

Interpretative listening: … is founded on an awareness that an activeinterpretation – a sort of reaching out rather than taking in – isinvolved, whereby the listening is deliberate and aware of the falli-bility of the sense being made.

(ibid. pp. 52–3)

It is worth noting that the more involved and committed the teacher, theharder it is to maintain awareness of possible misinterpretation. These two


modes are founded on a fundamental distinction between teacher andlearner.

Hermeneutic listening: … is more negotiatory, engaging, messy, involvingthe hearer and the heard in a shared project [it] is an imaginative participa-tion in the formation and transformation of experience through anongoing interpretation of the taken-for-granted and the prejudices thatframe perceptions and actions.

(ibid., p. 53)

Listening in order to teach: Teacher report

After being involved in a year-long research project regarding listening, oneteacher reported:

I have become a better listener. Teachers are basically talkers who feel astrong desire to share their knowledge with other people. Children areno different. If we really make an effort to listen to our students, we willbecome the richer for it.

(quoted in Cobb, Wood and Yackel, 1990, p. 135)

Subordinating teaching to learning: Caleb Gattegno

According to Gattegno, there are four tasks facing a teacher who wants tosubordinate teaching to learning. In the following extracts, Gattegno had inmind a wide variety of situations, such as: things have multiple names; senseimpressions act upon us and we upon them; we use direct experience fortesting what we hear from others.

… to become a person who knows himself and others, as persons. This isno mere sentimental homily, but means that the teacher must recognizethat beyond any individual’s behaviors is a will which changes behaviorsand integrates them.

(Gattegno, 1970, p. 53)

The second task of the teacher is to acknowledge the existence of asense of truth which guides us all and is the basis of all our knowing.

(ibid., p. 56)

… to find out how knowing becomes knowledge.Since this problem applies to himself as well as to all others, what he

needs to do is to watch himself making this transformation. …(ibid., p. 60)


… the duty to consider the [principle of] economy of learning. …[ … ]A reflection on the acts of living will show us that to live is to change

time into experience. So time must be considered as what we areendowed with by the act of coming into the world and that consumptionof time, if it is not to be destructive for the individual, should lead tosome equivalent worth in terms of experience, which when accumu-lated, becomes growth.

(ibid., pp. 63–4)

The role of the teacher of mathematics is to recognize that a student whocan speak has a large number of mental structures which can serve asthe basis for awarenesses that will enable him to transform these struc-tures into mathematical ones.

(ibid., p. 70)

Gattegno stressed the need for listening as a form of teaching – one of theinstruments that teachers can use to inform themselves about what learnersare doing and thinking (see p. 225). See also conjecturing atmosphere,p. 139, Maturana and Varela, p. 70, and Warden, p. 277.


Teaching can be seen as instructing, as guiding, as interacting, as fostering andsustaining mathematical thinking, even as listening, and it can be seen throughthe lenses of various metaphors (see Fox, p. 42). Listening can be evaluative orinterpretative.

9 Initiating mathematicalactivity Initiating mathematical activity

Introduction

In this chapter the focus is on getting mathematical activity started. Thechapter starts out with some principles, both philosophical and pragmatic. Itthen goes on to consider the practicalities of designing tasks, and selectingcontexts and apparatus for learning.

Principles

The principles considered in the extracts in this section include the desirabliityof starting teaching with a question for learners to work on, the learners under-standing the purpose of the mathematics behind the activity, and working fromwhat learners already know and understand or have experienced.

Starting with a question: Paul Halmos

Paul Halmos (b. Hungary, 1916–) moved to the USA when he was eight. Hebecame an eminent mathematician and is widely acknowledged as a bril-liant expositor. He published an ‘automathography’ and has written manyarticles on the art of teaching. He was inspired by the teaching style of R. L.Moore (1882–1974), who provided graduate students with a list of theo-rems and required them to find the proofs, having agreed not to talk toeach other outside of class or to consult textbooks. Most of the manyfamous mathematicians who went through Moore’s class thought it was thebest they had ever taken, though some found it inimical to their preferredways of working.

Let me emphasize one thing … . The way to begin all teaching is with aquestion. I try to remember that precept every time I begin to teach acourse, and I try even to remember it every time I stand up to give alecture … .

Another part of the idea of the method is to concentrate attention onthe definite, the concrete, the specific. Once a student understands,

really and truly understands, why 3 × 5 is the same as 5 × 3, then hequickly gets the automatic but nevertheless exciting and obvious convic-tion that ‘it goes the same way’ for all other numbers.

(Halmos, 1994, p. 852)

Note the similarities with making use of learners’ powers to generalise (seep. 137 and p. 196). The purpose of an example is to afford access to a wholespace of possibilities, not to focus attention on a particular example.

Halmos was convinced that learners have to come to see ‘why’ things aretrue for themselves (see also teaching for understanding, p. 293):

What we can do is to point a student in the right directions, challengehim with problems, and thus make it possible for him to ‘remember’ thesolutions. Once the solutions start being produced, we can comment onthem, we can connect them with others, and we can encourage theirgeneralizations. The worst we can do is to give polished lecturescrammed full of the latest news from fat and expensive scholarly jour-nals and books – that is, I am convinced, a waste of time.

(ibid., p. 851)

For Halmos, the most a teacher can do is create conditions and provide waysof working on mathematics. Most attempts at teaching end up interferingwith learning, which resonates with Gattegno’s booktitle: What We OweChildren: The subordination of teaching to learning.

Here are more observations from Halmos about teaching, which areconsistent with his view that mathematics is centrally about problem solving(see p. 187).

The best way to learn is to do; the worst way to teach is to talk.[ … ]A famous dictum of Polya’s about problem solving is that if you can’t

solve the problem, then there is an easier problem that you [can] solve –find it! If you can teach that dictum to your students, teach it so that thatthey can teach it to theirs, you have solved the problem of creatingteachers of problem solving. The hardest part of answering questions is toask them; our job as teachers and teachers of teachers is to teach how toask questions. It’s easy to teach an engineer to use a differential equationscook book; what’s hard is to teach him (and his teacher) what to do whenthe answer is not in the cook book. In that case, again, the chief problemis likely to be ‘what is the problem?’. Find the right question to ask, andyou’re a long way toward solving the problem you’re working on.

(Halmos, 1975, pp. 466–7)

Telling people things is nevertheless the most common form of instruction.Constructivism (see p. 54) was used to advocate movement away from

Initiating mathematical activity 229

lecturing towards investigations and discussion in order that learners would besupported in ‘making sense’, in ‘constructing meaning’. But the constructivistnotion that learners have to make their own sense, or to be enculturated intothe practices (particularly the linguistic ones) of the community appliesequally well to learners sitting in rows in lectures and learners exploring forthemselves or engaging in scientific debate. What matters is whether thelearners are in a position to be able to see, hear and make sense of what theyare told and shown. Montaigne caught this point beautifully, if ornately:

If only Nature would deign to open her breast one day and show us themeans and the workings of her movements as they really are (firstpreparing our eyes to see them).

(Montaigne, 1588, p. 602)

Montaigne went on to quote Plato as saying that ‘nature is but enigmaticpoetry’, and that Nature intends to exercise our ingenuity. Our relationshipto Nature through science is more developed since Montaigne’s time thanperhaps our relationship with (human) psyche. Preparing a learner to beable to see and hear is what Dewey was getting at in his phrasepsychologising the subject matter (see also p. 45).

Psychologising the subject matter: John Dewey

Dewey argued strongly that effective education takes account of the inter-ests, concerns, powers and potential of learners, and transforms the subjectmatter so that it is appropriate to the learner. Dewey called thispsychologising the subject matter. Lee Shulman (see p. 41) adapted and builton this when he distinguished pedagogic subject matter as one of the typesof knowledge required to be a teacher.

Here Dewey argues against the approach to curriculum design simply aslaying out the subject matter clearly and logically.

A psychological statement of experience follows its actual growth; it ishistoric; it notes steps actually taken, the uncertain and tortuous, as well asthe efficient and successful. The logical point of view, on the other hand,assumes that the development has reached a certain positive stage offulfilment. It neglects the process and considers the outcome. It summa-rizes and arranges, and thus separates the achieved results from the actualsteps by which they were forthcoming in the first instance. We maycompare the difference between the logical and the psychological to thedifference between the notes which an explorer makes in a new country,blazing a trail and finding his way along as best he may, and the finishedmap that is constructed after the country has been thoroughly explored.The two are mutually dependent. Without the more or less accidental anddevious paths traced by the explorer there would be no facts which could


be utilized in the making of the complete and related chart. But no onewould get the benefit of the explorer’s trip if it was not compared andchecked up with similar wanderings undertaken by others; unless thenew geographical facts learned, the streams crossed, the mountainsclimbed, etc., were viewed, not as mere incidents in the journey of theparticular traveler, but (quite apart from the individual explorer’s life) inrelation to other similar facts already known. The map orders individualexperiences, connecting them with one another irrespective of the localand temporal circumstances and accidents of their original discovery.

[ … ]… But the map, a summary, an arranged and orderly view of previous

experiences, serves as a guide to future experience; it gives direction; itfacilitates control; it economizes effort, preventing useless wandering,and pointing out the paths which lead most quickly and most certainlyto a desired result. Through the map every new traveler may get for hisown journey the benefits of the results of others’ explorations withoutthe waste of energy and loss of time involved in their wanderings –wanderings which he himself would be obliged to repeat were it not forjust the assistance of the objective and generalized record of theirperformances. …

[ … ]Hence the need of reinstating into experience the subject-matter of

the studies, or branches of learning. It must be restored to the experi-ence from which it has been abstracted. It needs to be psychologized;turned over, translated into the immediate and individual experiencingwithin which it has its origin and significance.

… The problem of the teacher is … concerned with the subject-matterof the science as representing a given stage and phase of the developmentof experience. His problem is that of inducing a vital and personal expe-riencing. Hence, what concerns him, as teacher, is the ways in whichthat subject may become a part of experience; what there is in the child’spresent that is usable with reference to it; how such elements are to beused; how his own knowledge of the subject-matter may assist in inter-preting the child’s needs and doings, and determine the medium inwhich the child should be placed in order that his growth may be prop-erly directed. He is concerned, not with the subject-matter as such, butwith the subject-matter as a related factor in a total and growing experi-ence. Thus to see it is to psychologize it.

(Dewey, 1902, pp. 19–23)

There are similarities with educating awareness (see p. 61 and p. 161) andwith didactical phenomenology (see p. 202).

Dewey went on to criticise educational approaches that ignore the experi-ence of the learner, and so fail to psychologise the subject matter.


It is the failure to keep in mind the double aspect of subject-matterwhich causes the curriculum and child to be set over against each other… . The subject-matter, just as it is for the scientist, has no direct relation-ship to the child’s present experience. It stands outside of it. The dangerhere is not a merely theoretical one. We are practically threatened on allsides. Textbook and teacher vie with each other in presenting to thechild the subject-matter as it stands to the specialist. Such modificationand revision as it undergoes are a mere elimination of certain scientificdifficulties, and the general reduction to a lower intellectual level. Thematerial is not translated into life-terms, but is directly offered as a substi-tute for, or an external annex to, the child’s present life.

(ibid., pp. 23–4)

There are similarities with attacks by Spencer on rote learning and rule teaching(see p. 152 and p. 196), and by Whitehead on inert knowledge (see p. 288).

Dewey then considered the implications of the failure to psychologise,particularly concerning too rapid use of symbols for experiences (whatmight also be referred to as awarenesses, see p. 61).

Three typical evils result: In the first place, the lack of any organicconnection with what the child has already seen and felt and lovedmakes the material purely formal and symbolic. There is a sense inwhich it is impossible to value too highly the formal and the symbolic.The genuine form, the real symbol, serve as methods in the holding anddiscovery of truth. … But this happens only when the symbol reallysymbolizes – when it stands for and sums up in shorthand actual experi-ences which the individual has already gone through. A symbol which isinduced from without, which has not been led up to in preliminary activ-ities, is, as we say, a bare or mere symbol; it is dead and barren. …

The second evil … is lack of motivation. There are not only no facts ortruths which have been previously felt … with which to appropriate andassimilate the new, but there is no craving, no need, no demand. Whenthe subject-matter has been psychologized, that is, viewed as an out-growth of present tendencies and activities, it is easy to locate in thepresent some obstacle, intellectual, practical, or ethical, which can behandled more adequately if the truth in question be mastered. This needsupplies motive for the learning. An end which is the child’s own carrieshim on to possess the means of its accomplishment. But when material isdirectly supplied in the form of a lesson to be learned as a lesson, theconnecting links of need and aim are conspicuous for their absence.

(ibid., pp. 24–5)

There are similarities with the transposition didactique (see p. 83), and the role ofobstacles (see p. 303) and disturbance (see p. 55).


The third evil is that even the most scientific matter, arranged in mostlogical fashion, loses this quality, when presented in external, ready-made fashion, by the time it gets to the child. It has to undergo somemodification in order to shut out some phases too hard to grasp, and toreduce some of the attendant difficulties. What happens? … The reallythought-provoking character is obscured, and the organizing functiondisappears. Or, as we commonly say, the child’s reasoning powers, thefaculty of abstraction and generalization, are not adequatelydeveloped.

(ibid., p. 26)

Note the reference to learners’ powers (see p. 115 and p. 233). The principal roleof the curriculum developer and the teacher is to psychologise the subject matter.The Structure-of-a-topic framework (see p. 203) can be helpful in this regard.

Finally, Dewey returned to learners’ powers.

It is his present powers which are to assert themselves; his present capaci-ties which are to be exercised; his present attitudes which are to be real-ized. But save as the teacher knows, knows wisely and thoroughly, therace-expression which is embodied in that thing we call the Curriculum,the teacher knows neither what the present power, capacity, or attitude is,nor yet how it is to be asserted, exercised, and realized.

(ibid., p. 31)

This passage captures the essence of most attempts to reform education. It wasa rallying cry for what in the 1970s became the child-centred curriculum, itselfcriticised for leading to practices that allowed children to proceed at their ownpace without making full use of their powers. It summarises the key constructsthat lie at the heart of effective teaching and successful learning.

Diagnostic teaching: Alan Bell

Alan Bell (b. Kent, 1929–) has been researching mathematics classrooms formany years as a member of the influential Shell Centre at the University ofNottingham.

The Diagnostic Teaching project began by identifying a multitude ofcommon errors and misconceptions underlying those errors in arithmetic,and problems which proved difficult because learners often choose an inap-propriate operation. Teaching material was then prepared, specificallytargeted at ‘notational misconceptions, numerical misconceptions, and theinvariance of quantity relations in problems in contexts such as price andspeed’ (Bell, 1986, p. 26).

By invariance here, the author meant, for example, that (average) distancealways equals (average) speed multiplied by time.


… a short start activity … would lead to the exposure of such miscon-ceptions as are present in the pupils’ schemes and ideas. So it could besaid that we deliberately gave them questions which at least some pupilswould get wrong; and without forewarning them of possible hazards.The principle was that if there is an underlying misconception, then it’s‘better out than in’; it needs to be seen and subjected to critical peergroup discussion. Of course, establishing a classroom atmosphere inwhich this is an accepted activity is not a trivial matter, and it may takesome time.

[ … ]… The working assumption was that if the difficulties with notation

and the numerical misconceptions could be overcome, then correctchoices of operation would be made. …

… we saw the ‘conflict-discussion’ as the main learning experience,and the written-work as introductory, giving the opportunity for openingup the situation and allowing some mistakes to be made which led toconflicts and hence the discussion. Following the discussions we gavesome written work to ‘consolidate’ the understanding gained. Thisconsisted of similar problems but with feedback enabling immediatecorrection to be made if necessary.

… Perhaps the most striking observation from all this work is thatback-sliding is the norm. Even after clearly effective lessons withlearning visibly taking place, in the next lesson most of the class couldslip again into the original error. True, the second recovery was quickerthan the first. The method of conflict-discussion promises to provide amore effective way of dealing with this widely recognised phenomenonthan simply reteaching. The same key conceptual points do need to bethe focus of discussion repeatedly; but they need to appear in differentcontexts and modes of presentation.

(ibid., pp. 27–9)

The authors’ first attempts produced tasks that proved to be too rigid. Butwhat did develop was discussion around various alternatives. They endedup with a chart with cards containing numbers, statements in wordsconcerning operations, and symbolic representations of those operations.Groups of learners then sorted the cards onto a chart, seeking agreement asto where cards belonged.

In a follow-up article (there were to be a further three articles), Bell reportedon getting learners to make up their own questions and to mark and commenton the homework of a fictitious learner who made many classic mistakes.

In general these are hard tasks; though the difficulty decreases withfamiliarity. Our conclusion from our research has been that this is what isrequired to provoke worthwhile discussion and to strengthenunderstanding. …


[ … ]Students’ early attempts to make up questions showed a surprising

level of difficulty. Frequently the questions were unanswerable. Some-times they simply gave data but asked no question; in other cases theygave insufficient information. …

In later attempts students gave more coherent questions, and theseshowed up some important misconceptions [especially when they gavetheir own answers!].

[ … ]In another form of making up questions, … students were given a set

of related quantities, but just one numerical value, and were asked towrite two possible questions, one containing easy numbers, and onewith hard numbers. This gave the students the challenge of recognisingwhat were ‘hard numbers’ in these problems, as well as having tooperate with them.

These tasks offer a variety of different demands and constraints; eachhas some element of freedom for creativity, and also some means ofensuring that the more difficult number and quantity combinations arefaced as well as the easier ones.

(Bell, 1987, pp. 21–2)

Principles: Alan Bell

Bell subsequently considered how activity can be turned into learning:

Most uses of mathematics involve a cycle of mathematization, manipula-tion, and interpretation – that is, recognizing in the given situation therelevance of some mathematical relationship, expressing this relationsymbolically, manipulating the symbolic expression to reveal some newaspect, and interpreting this new aspect or giving some fresh insight intoit in the given situation. …

[ … ]Given that the pupils are to be offered activities which embody the

characteristic mathematical strategies, and which embrace the majorconcepts of the subject, what needs to be done to turn these intolearning experiences? That is, what can we do to make it more likelythat the pupils will actually perform better when they meet these orsimilar tasks again? Learning is not just success in the present task butimprovement in capability. This factor has been neglected in somerecent pedagogies, which assume that a sequence of gently gradedproblem-solving tasks results in learning.

… to develop the strategy of generalizing one would offer a set of expe-riences, of different types, in different contexts and in different conceptualfields, but each requiring the forming and expressing of some generaliza-tion, and one would draw attention to the characteristic features of the


process. Particular skills or concepts would be dealt with in a similar way.It is thus appropriate to alternate general exploratory activities with workfocused on related specific concepts, strategies, or skills.

In some respects, the requirements conflict. For example, in anactivity aimed at developing the ability to carry through an investigationin which one follows up each discovery by choosing an appropriatequestion to tackle next, one cannot control which concepts and skillswill be involved in the work as it progresses. Conversely, when the aimis to work on some particular concepts and skills, it is necessary for thediscussion to be guided so as to explore the various aspects of thoseconcepts; one cannot at the same time allow the inquiry to take its directionfrom what appears the most relevant question to ask next.

(Bell, 1993, pp. 6–8)

Bell then offered some explicit principles for designing teaching:

… First one chooses a situation which embodies, in some contexts, theconcepts and relations of the conceptual field in which it is desired towork. Within the situation, tasks are proposed to the learners whichbring into play the concepts and relations. It is necessary that the learnershall know when the task is correctly performed; hence some form offeedback is required. When errors occur, arising from some misconcep-tion, it is appropriate to expose the cognitive conflict and to help thelearner to achieve a resolution. This is one type of intervention whichthe teacher may make to assist the learning process.

(ibid., pp. 8–9)

Note how Bell made reference in the passage to many of the ideas presentedelsewhere in this reader.

Another general mode of intervention is in adjusting the degree of chal-lenge offered to the learner by the task; the extent to which the task itselfprovides this flexibility is a significant task feature. The next requirementis for ways of developing a single starting task into a multiple task,bringing the learner to experience a rich variety of relations within thefield. Typically, this can be done by making changes of element (e.g., typeof number), structure, and context. The degree of intensity of this complexof learning experiences is an important factor. Reflection and review areother key principles; they imply the perception and study not only of thebasic concepts and relations within the tasks but also of the properties ofthe different types of problem within the field and of the methods of solutionfound – meta-knowledge of the tasks and of the activity.

(ibid., pp. 8–9)

There are similarities with dimensions-of-possible-variation (see p. 56).


A fundamental fact about learned material is that richly connected bodiesof knowledge are well retained; isolated elements are quickly lost.

(ibid., p. 9)

Compare this with situated cognition (see p. 86), rote learning (see p. 151),and knowing (p. 289) and understanding (p. 293).

In conflict-discussion lessons … feedback is an integral part of theprocess of discussion … . In other tasks, including games, the mode ispredict and check … .

Refection and Review: Exploring relationships and resolving conflictsthrough discussion are in themselves reflective activities. Here we implysomething more – a more global reflection on the process of performingthe task and identifying the crucial steps, and on the new knowledgegained and how it fits into one’s existing body of knowledge. This devel-opment of awareness is important in labelling the newly gained knowl-edge in memory in such a way as to make it accessible on relevant futureoccasions.

[ … ]Intensity: It is well known that repetition is an essential element in

learning. Questions remain concerning the effects of the degree ofvariety in the set of tasks; some of these are implicit in the foregoingdiscussion. There is clearly no general answer to this question, but thereis evidence in some experiments that what might be regarded as exces-sive repetition has resulted in striking gains. It is also clear that whilerepeated memorization tasks may produce short-lived results, intensiveinsight-demanding tasks produce longer-term gains.

[ … ]Feedback: In some cases feedback is intrinsic to the task; in other

cases, a predict-and-check-mode may be possible – for example, amental calculation may be compared with a calculator result. In somegames, the opponent may challenge a doubtful answer. … Sometimesdefinite feedback of correctness is difficult or impossible, but provisioncan be made for discussion of the task in pairs, or in a group, so that atleast some errors and misconceptions can be detected.

(ibid., pp. 15–19)

Principles: Jan van den Brink

Van den Brink took the principles suggested by Alan Bell and colleagues,and illustrated some while challenging others by using examples.

• Richly connected bodies of knowledge are well retained;• Discussion of a few hard critical problems is more effective than prog-

ress through a sequence of many easy questions;


• Pure practice increases fluency but does not develop understanding( … our pupils’ demonstrated that striking surprises are important fordeveloping understanding);

• Scope for pupil choice and creative productions can provide bothmotivation and challenge at the pupils’ own level;

• Establishment of multiple connections is helped by exploring fullythe relationships in one context before moving to another context (I[van den Brink] do not agree with this principle. … There ought to bemerely a mutual influence between formal contexts and daily ones.)

(van den Brink, 1993, pp. 62–3)

The age-old disagreement about relevance and the role of familiar contextappears again (see also p. 82, p. 110, p. 264 and p. 292).

Van den Brink also added two more principles:

• Correcting the unexpected conflicts and surprises (mostly oppositesof well-known relations) is powerful learning.

• Criticizing by teachers improves teaching … the role of the teachers incriticizing [draft] textbooks improved the teaching material.

(ibid., p. 63)

Tasks

In this section, tasks are considered as devices for intiating activity. Tasksare drawn, usually by the teacher from a task-space full of possible varia-tion. The section includes a classification of types of task and a consider-ation of the possibility of the learner becoming aware of the dimensions-of-possible-variation.

Structure of tasks

Task and activity: Bent Christiansen and G. Walther

Christiansen and Walther drew on Vygotsky’s notion of activity to distin-guish between a task that is proposed and the activity that may arise fromengaging in (some version of) that task. See also teaching dilemma (p. 96).

The tasks and the activities establish so to speak the ‘meeting place’between teacher and learner.

(Christiansen and Walther, 1986, p. 246)


Psychologising the subject matter is what teachers can do for learners: estab-lishing an environment in which they encounter cognitive conflicts.

… students – by means of tasks set by the teacher – may be initiatedinto an appropriate spectrum of mathematical activity. However, anumber of teacher-actions are needed in each case to ensure that theeducational activity in question results in learning as intended. … anyactivity proceeds through goal directed actions which are ‘inherent’ in,but not ‘given by’ the task; … specific tasks are needed to motivatespecific types of activity (e.g. of exploratory or problem-solving types);… any activity contributes to learning of different types and at differentcognitive levels; … specific teacher-actions are needed to ensure thatpersonal knowledge is developed in an appropriate degree into sharedknowledge.

(ibid., pp. 253–4)

Learners have to interpret any task, and will naturally do so in terms ofwhat they can imagine themselves doing. Alternatively, they may waituntil the task is specified in such detail that they can undertake it butwithout having to engage, which amounts to a form of funnelling (seep. 274).

… even when students work on assigned tasks supported by carefullyestablished educational contexts and by corresponding teacher-actions,learning as intended does not follow automatically from their activity onthe tasks. …

[A learner’s engagement with a task is influenced by] the individual’sinterest in the task, his motivation for acting, his attitudes towards theteacher and the school, his conceptions of learning and of mathe-matics. And … whether he reflects on his actions and on his ownlearning.

(ibid., p. 262)

Task design: Konrad Krainer

Konrad Krainer is an academic at the University of Klagenfurt, in Austria. Hisinterests span educational management and mathematics education. Here heraises the endemic tensions facing any teacher or curriculum developer.

How should mathematics instruction be organized? This is one of themost important questions to be dealt with in mathematics education.There are two extreme answers to this question.

The first sees mathematics as a highly complex and highly developedscience which offers, however, polished and stable ideas and theories inareas understandable for pupils … . Therefore, it is easy to build up well-established (‘secured’) courses for mathematics … . According to thesecond answer to this question, pupils bring a variety of relevant prac-tical experiences, associations, intuitions, and so on to mathematics


instruction. If the spontaneity and creativity of the pupils are taken seri-ously it is – from a psychological point of view – necessary to have acertain insecurity of mathematics courses … .

[ … ]This dilemma cannot be resolved by a didactical theory. The situation

remains one of conflict because both extremes embody meaningfuldemands: on the one hand, the demand for economical efficiency andfor well-developed ‘motorways’, and on the other hand, the demand thatthe pupils should investigate and discover for themselves and have thefreedom to ‘pave’ their own ways.

(Krainer, 1993, pp. 66–7)

Compare this with the didactic tension emerging from the contrat didactique(Brousseau, p. 79).

Krainer applied his more general perspective to the design of ‘powerful’mathematical tasks, identifying various properties that should be incorpo-rated in the tasks.

1a Team spirit: … [the] tasks should be well interconnected with othertasks. The ‘horizontal’ connection of tasks can be seen as a contribu-tion to the security of mathematics courses.

1b Self-dynamics: … tasks facilitate the generation of further interestingquestions. The ‘vertical’ extension of tasks to open situations can beseen as a contribution to the insecurity of mathematics courses.

Powerful tasks therefore embody the dilemma security–insecurityas a constituting element.

2a High level of acting: … the initiation of active processes of conceptformation which are accompanied by relevant (‘concept generating’)actions.

2b High level of reflecting: … acting and reflecting should always beseen as closely linked. An important aspect of reflection refers tofurther questions from the learners (which in their turn could lead tonew actions).

These two [sets of] properties express the philosophy that learnersshould be seen not only as consumers but also as producers of knowl-edge. The teacher’s task is to organize an active confrontation of thepupils with mathematics. Powerful tasks are important points of contactbetween the actions of the teacher and those of the learner.

(ibid., p. 68)

Here Krainer is advocating thinking of tasks as systems of tasks, nevercomplete, but indicating directions of possible development and chal-lenge. This fits with the notion of tasks as samples from a task-domain ortask-space in which individual tasks arise as choices made in a variety of


dimensions-of-possible-variation, within a range-of-permissible-change(see p. 57). The task-domain consists of all possible variations, and it is tothe advantage of the learner as well as the teacher to become aware ofthat space. Indeed, each individual will have their own version of a task-space, consisting of the dimensions-of-possible-variation and the range-of-permissible-change within each of those dimensions of which they areaware.

Framing tasks: Paolo Boero

Paolo Boero (b. Italy, 1941–) is a leading mathematics educator andresearcher, known for his detailed research into the design and analysis oftasks for use in schools. Boero used the notion of a field of experience to gobeyond mere context, appealing to what learners already have experiencedand from which rich mathematical ideas can emerge:

The field of experience of sunshadows is a context in which studentscan naturally explore problem situations in different dynamical ways. Inorder to study the relationships between sun, shadow and the objectwhich produces the shadow, one can imagine (and, if necessary,perform a concrete simulation of) the movement of the sun, of theobserver and of the objects which produce the shadows.

The field of experience of sunshadows was chosen because it offersthe possibility of producing, in open problem solving situations, conjec-tures which are meaningful from a space geometry point of view, noteasy to be proved and without the possibility of substituting proof withthe realization of drawings.

(Boero et al., 1996 webref)

Outer and inner aspects of tasks: Dick Tahta and John Mason

From his work with learners on watching and making mathematical anima-tions, Tahta drew attention to a distinction between the overt or outer taskwhich is set, and the inner aspects which are expected to emerge throughactivity arising from that task. This notion can be developed even further toinclude meta-aspects of tasks:

Tahta (1980; 1981) distinguished inner and outer meaning of tasks,and to these I add meta meaning. Outer meanings have to do withexplicit content such as known mathematical results described interms of a mathematical label which purports to summarise a mathe-matical story. To exploit the outer meaning of a task, it makes sense toengage students in story telling by reconstructing what they haveseen by giving brief-but-vivid accounts of fragments from the task


activity, and weaving that into a story which accounts for those frag-ments. Throughout this story weaving, negotiation with others plays acrucial role.

Inner meaning refers to global awareness such as multiplicity of defi-nitions or of perspective, the linking together of previously disparateelements into one continuous family, the perception of an infinite classof elements as a single entity, the choosing of constraints and the effectof those constraints, the stressing of different points of view that yield aninvariant result, the simultaneous holding of several points of view bymeans of one invariant. …

Meta meaning refers to the opportunities provided by any mathemat-ical exercise to observe one’s own behaviour and propensities, andthereby perhaps to increase sensitivity so as to inform action in thefuture.

(Mason, 1992, p. 12)

Task design: David Wood, Jerome Bruner and Gail Ross

David Wood and his colleagues were designing research tasks for youngchildren in order to explore the ways in which mothers supported or tutoredtheir children in the face of difficulties.

The task set the children was designed with several objectives in mind.First and foremost, it had to be both entertaining and challenging to thechild while also proving sufficiently complex to ensure that his behav-iour over time could develop and change. It had to be ‘feature rich’ inthe sense of possessing a variety of relevant components. We tried tomake its underlying structure repetitive so that experience at one pointin task mastery could potentially be applied to later activity, and a childcould benefit from after-the-fact knowledge or hindsight. But the taskhad not to be so difficult as to lie completely beyond the capability ofany of the children. And finally, we did not want to make too greatdemands upon the child’s manipulatory skills and sheer physicalstrength.

(Wood, Bruner and Ross, 1976, p. 91)

See also p. 266 and p. 268 for principles Wood et al. developed from theirstudies concerning tutoring, especially scaffolding.

Task types

There are many different ways of classifying types of tasks. They can be seenin terms of:


• A phenomenon to be explored and explained (in the material worldoutside school, in common experience, on an electronic screen, inmental imagery, in symbols);

• Instructions to follow on self-chosen objects which contribute to collec-tive data that then become a phenomenon to be explained;

• Routine exercises which can be opened out in various ways;• Invitations to construct objects meeting various constraints, as a step

towards appreciating a general class of objects rather than simply partic-ular cases;

• Invitations to construct problems for the learners’ peers, the teacher, orother adults, in order to encourage thinking about the class of ‘problemslike this’ and the technique for resolving them, as a step towards appre-ciating a general class of problem.

This perspective stresses the source of something on which to work. Taskscan also be classified in terms of what learners are invited to do. Forexample, use data obtained from the Internet, from a book or self-collected;seek advice in response to questions posed by other (possibly imaginary)learners; sort or classify objects, statements or expressions; construct phys-ical, mental or symbolic objects; interpret diagrams, graphs, and expressions;explore the significance of some way of doing things such as a scoring mech-anism in some sport.

Task dimensions: Doug Clarke

A variety of task dimensions have been studied and used by researchers andcurriculum developers. Working in Australia, Clarke and his colleagues(Clarke, webref) have developed criteria for rich assessment tasks, includingthat: the tasks connect with and arise naturally from the topics to be taught,they are manageable in the time; they have multiple outcomes and aresusceptible to multiple approaches; the way of working encourages groupwork and communication as well as individual work; they authenticallyrepresent use of skills and knowledge in the future.

Like Clarke, Higgins also made use of dimensions concerning rich orimpoverished tasks laid out by Steve Leinwand and Grant Wiggins (seeStenmark, 1991; Higgins, webref) by using distinctions that are often intension. Each row below is seen as a spectrum. It is not a matter of ruling outtasks that fail to meet the requirements of the left-hand end of each distinc-tion, but rather to use these as ideals to be approached along a spectrumfrom right to left in as many categories as possible.


(Adapted from Higgins, webref)

The terms used for these distinctions display a strong bias and commit-ment to tasks used for assessment purposes. Further dimensions are neededfor tasks that are designed to work on, develop and probe facility withroutine techniques, and for tasks designed to expose learners to newconcepts or to encounter pervasive themes (see p. 193) or the developmentof powers (see p. 115). See also Ahmed (1987). Although tasks are often clas-sified as either open or closed as in the last row of the table, (or even open-ended or open-fronted), it is not the task that is open or closed, but thepeople who work on it who open it out or close it down. What matters is notso much the type of the task, as the activity which arises from it and thepossibilities afforded.

Opening up tasks: Sherman Stein

Stein, who wrote about how algorithms drive out thought (see p. 177), wenton to advocate ‘open field’ tasks.

What is the point of … [an] exercise? Is it to check a definition [is under-stood] or a theorem or the execution of an algorithm? … Blinders areplaced on the student to focus attention on particular facts or skills. For


ESSENTIAL Fits into the core of the curriculumRepresents a ‘big idea’

TANGENTIAL

AUTHENTIC Uses processes appropriate to the disciplineLearners value outcome of process

CONTRIVED

RICH Leads to other problemsRaises other questions

Has multiple possibilities

SUPERFICIAL

ENGAGING Thought-provoking;fosters persistence

UNINTERESTING

ACTIVE Learner is worker and decision maker;learners interact with other learners;

learners construct meaning and deepenunderstanding

PASSIVE

FEASIBLE Can be done within school andhomework time;

developmentally appropriate forlearners; safe

INFEASIBLE

EQUITABLE Develops thinking in a variety of styles;contributes to positive attitudes

INEQUITABLE

OPEN Has more than one right answer;has multiple avenues of approach making it

accessible to all learners

CLOSED

instance we ask students to factor x4 – 1. An open field exercise puts noblinders on the student. We might ask ‘for which positive integers n doesx2 – 1 divide xn – 1?’ An open-field exercise may not connect with thesection covered that day; it may not even be related to the course. Suchan exercise may require the student to devise experiments, make aconjecture, and prove it.

(Stein, 1987, pp. 3–4)

The notion of dimensions-of-possible-variation (p. 57) provides a languagefor discussing with learners how they can open up any task for themselves:what things could be varied, yet the task remains much the same?

Situations and apparatus

The use of situations which arise as a context for mathematics has beenadvocated by many authors with different justifications (see authenticactivity, p. 108, real problem solving, p. 114, realistic mathematics, p. 110);furthermore, the use of physical apparatus to create situations and to supportthe learning of mathematics has been advocated since ancient times (seePlato p. 34). It transpires that the use of apparatus, or cultural tools (see p. 85)is highly problematic. The extracts in this section explore this issue.

Making use of situations

For an expert, mathematics appears to be embedded within the apparatus,but a learner has to distinguish what is mathematical from what is happen-stance and particular to the situation involving the apparatus. In a very realsense, the concept has to be understood, at least to some extent, in order toappreciate how the apparatus displays or reveals the mathematical ideas.

… when a teacher presents a child with some apparatus or materials …he [or she] typically has in mind some one particular conception of whathe [or she] presents in this way. But then the incredible assumptionseems to be made that the teacher’s conception of the situation


Distinguishing tasks as set (and as conceived by author and by teacher) fromactivity arising from work on those tasks (as construed by learners) helps tofree teachers from expectations that learners will all do the same thing in thesame way, and to open up sensitivity to the richness of what emerges. This willalso facilitate awareness of the kinds of questions to ask which might promoteparticular mathematical awarenesses and thinking. Seeing tasks as having innerand meta-aspects, as well as outer overt form, opens up thinking about tasks inrelation to the Structure-of-a-topic framework. Tasks can arise from a field ofexperience rather than from a simple context.

somehow confers a special uniqueness on it such that the children mustalso quite inevitably conceive of it in this way too.

(Dearden, 1967, pp. 145–6,quoted in Cobb, Yackel, and Wood, 1992, p. 9)

Affordances: James Gibson and James Greeno

James Gibson (Ohio, 1904–1979) was an American psychologist particularlyinterested in perception. He coined the term affordance to refer to thecomplex interrelationship between animal and environment.

The affordances of the environment are what it offers the animal, what itprovides or furnishes, either for good or ill. The verb ‘to afford’ is foundin the dictionary, but the noun ‘affordance’ is not. I have made it up. Imean by it something that refers to both the environment and the animalin a way that no existing term does. It implies the complementarity of theanimal and the environment …

(Gibson, 1979, p. 127)

James Greeno (b. South Dakota, 1935–) is an American sociologist-psychologistand mathematics educator. He developed Gibson’s notion of affordanceswithin mathematics education, which for him involved shifting attentionfrom individuals to interactions:

[In situativity theory] … cognitive processes are analyzed as relationsbetween agents and other systems. This theoretical shift does not implya denial of individual cognition as a theoretically important process. Itdoes, however, involve a shift of the level of primary focus of cognitiveanalyses from processes that can be attributed to individual agents tointeractive processes in which agents participate, cooperatively, withother agents and with the physical systems that they interact with.

[ … ]… In any interaction involving an agent with some other system, condi-

tions that enable the interaction include some properties of the agentalong with some properties of the other system. … The term affordancerefers to whatever it is about the environment that contributes to the kindof interaction that occurs. One also needs a term that refers to whatever itis about the agent that contributes to the kind of interaction that occurs.[Terms used include ability, effectivity, and aptitude.] …

Affordances and abilities, (or effectivities or aptitudes) are, in thisview, inherently relational. An affordance relates attributes of somethingin the environment to an interactive activity by an agent who has someability, and an ability relates attributes of an agent to an interactiveactivity with something in the environment that has some affordance.The relativity of affordances and abilities is fundamental. Neither an


affordance nor an ability is specifiable in the absence of specifying theother. It does not go far enough to say that an ability depends on thecontext of environmental characteristics, or that an affordance dependson the context of an agent’s characteristics. The concepts are codefining,and neither of them is coherent, absent the other, any more than thephysical concept of motion or frame of reference make sense withoutboth of them.

(Greeno, 1994, pp. 337–8)

Along with affordances there are constraints and attunements (what individ-uals are disposed or attuned to recognise as possibilities), including powers(see p. 115). Affordances are constrained as well as enabled by tools, rules,custom, language and power, so the actual possibilities are a subset of whatmight be possible. There is a complex interplay between what could bepossible, what is possible and what is seen as possible.

Play: Zoltan Dienes

Dienes strongly advocated exposing learners to multiple embodiments(several different situations in which the same idea arises) so that they canabstract the essence, just as three pens, three crayons, or three teddies areabstracted to ‘three’ quite naturally and spontaneously by young children:

Many people have suggested before … the natural way in which chil-dren acquire knowledge is through play in some form or another. It isnot at all clear, however, what the processes are that lead from play tothe purer forms of cognition such as construction of classifications,generalizations, logical classifications and deductions.

(Dienes, 1963, p. 21)

There are similarities with the role of intention in activity theory (see p. 84).Dienes then went on to distinguish three types of play: exploratory-

manipulative, which is curiosity-prompted play without specific aim ordirection; representational play in which the objects begin to stand for some-thing they are not, bringing imagination into ‘play’; and the search forregularities.

Manipulative play may quite imperceptibly move over to a search forregularities. When a rule or rules are found, play may occur which usesthese rules. Children delight in regularities, and the formulation of a rule-structure is a kind of closure which ties up all the loose ends of pastexperience. Children feel safe within such a rule-structure. Once it’sbeen thoroughly mastered, and has become part of the currency of play,the closure is often reopened by asking questions. In this case the rules


themselves will be the objects of manipulation and situation is openagain at a higher level.

(ibid., p. 23)

Dienes is making use of invariance (see p. 132 and p. 193), and there are simi-larities with modelling (see p. 190), and Polya’s three phases of exploring,formalising and assimilating (see p. 147).

Activity: Geoff Giles

Geoff Giles (b. Scotland, 1929–) was the founding director of the DIMEproject. He is a prolific generator of mathematical tasks and cultural tools, suchas angle measurers and tiles, for generating useful mathematical activity.

… The teacher provides the situation in which the activity takes place.The situation contained, explicitly or implicitly, a problem, i.e., the situa-tion provokes the child into activity. This activity is not to be effected bydirect intervention of the teacher and thus depends only on the initialsituation.

(Giles, 1966, p. 9)

Activity may not be mathematical. Operating a record player is not initself a musical activity although it may be closely connected. Similarly,playing with coloured rods might involve no mathematical activity.(They might represent soldiers, or flowers in a garden.) Mathematicsexists in the mind, and so mathematical activity is a mental activity. Wecannot observe it directly; we can only infer that it is taking place frombehaviour.

Activity may be mathematical and yet not desirable. Tackling aproblem involving two taps filling a bath with no plug is a mathematicalactivity but is undesirable.

(ibid., p. 10)

Appropriate tasks: Herbert Spencer

Spencer was an early advocate of learner exploration (see p. 116). He raisedthe question behind criticism of discovery learning (see p. 143):

If it be true that the mind, like the body … unfolds spontaneously – if itssuccessive desires for this or that kind of information arise when theseare severally required for its nutrition – if there thus exists in itself aprompter to the right species of activity at the right time – why interferein any way? Why not leave children wholly to the discipline of nature? …This is an awkward-looking question.

(Spencer, 1878, pp. 62–3)


Spencer drew attention to the way in which adults provide children withappropriate tasks and equipment, but the development is through a learner’sown use of their muscles and coordination. He proposed a parallel withintellectual development (see powers, p. 115).

… Thus, in providing from day to day the right kind of facts, prepared inthe right manner, and giving them in due abundance at appropriateintervals, there is as much scope for active ministration to a child’s mindas to its body.

(Spencer, 1878, p. 64)

Using situations: Colin Banwell, Ken Saunders and Dick Tahta

Banwell and his colleagues described the conditions they consider necessaryfor situations to be fruitful mathematically:

To be most fruitful, … situations should be able to be developed inmany different directions. They must be able to be initiated simply,immediately, and with a minimum demand upon existing vocabulary ortechnique. For there to be any creative response, they must be presentedwith flexible and experimental intent, often with deliberate ambiguity.

… working from situations … is not the same as working from prob-lems. Part of the activity is, in fact, the formulation of [local] problemsthat may arise out of the definitions and rules that are developed in thediscussion of the situation. Students will readily wish to solve problemsthat they have created themselves and such solutions are part of thework that follows from the starting point.

(Banwell, Saunders and Tahta, 1972, p. 66)

Many teachers have found that learner-generated tasks are more engagingthan worksheets and textbook tasks.

… In selecting, or recognizing, starting points that will be fruitful, theteacher may wish to see a [good] balance between those that are genu-inely open for himself as well as his students, and those that he knowsthrough experience or insight are very likely to lead to structures alreadyknown to him if not to students. In the latter case, it may be difficult forhim to restrain his influence on the choices that have to be made. We donot suggest that he should not make choices, but think it is important thathe should not only know when this is happening but be able to show it.

(ibid., p. 66)

When a teacher ‘knows’ what learners could encounter, the desire that thelearners do encounter it can turn into a strong temptation to make sure that


this happens (see p. 96). Banwell and his colleagues were advocating thatteachers be aware of, and be able to justify, the choices that they make.

Using situations: Janet Ainley

Ainley has been a primary school teacher, teacher educator, and researcher.She worked for some time with Skemp as a research fellow. Here she neatlycaptures a teaching dilemma between using situations as they arise, andsticking to a prepared plan. Using situations as they arise can be taken toextremes: either ignoring interests that have attracted learners’ attention, oron the other hand, being swayed and driven by whims in an attempt to makeand maintain contact with the learners’ interests.

Dead Birds: Pat likes to harness the children’s enthusiasms and interestsin the classroom, and tries to integrate work in all areas of the curric-ulum. Activities are rarely planned in advance, because this mightimpose adult interests on the children. Instead, all the work stems fromwhat the children bring to the classroom. Often this is in the form ofstories from their experiences at home, but occasionally they literallybring into school starting points from which work in a range of subjectsmay develop.

For example, one day a child found a dead bird on the way to school,and brought it to show to Pat and the rest of the class. Everything elsewas abandoned as the children became absorbed in studying this find.Lots of writing and artwork were inspired by the dead bird, but Pat wasalso pleased with the mathematical activities that the children did. Theyhad been doing some weighing activities earlier in the week, so some ofthem decided to weigh the bird, using a variety of informal units. Themaths table in the classroom was well equipped with measuring instru-ments, and other children wanted to use these to measure various partsof the bird. One group pulled out some of its feathers and timed themwith a stop-watch as they dropped to the ground. …

Easter Bunnies: Chris is also very concerned to link the mathematicsthat the children do to other curriculum areas. The class follow a mathe-matics scheme, but children work at their own pace through the text-books. The books are a bit dull, and Chris likes to make extra teachingmaterials for the class, following the progression in the textbook, butrelating mathematics to topics that the class are working on in otherareas.

Chris is a good artist, and makes attractive worksheets, which the chil-dren always enjoy using. Towards the end of the spring term last year,the class were preparing for Easter. In mathematics, many of the childrenwere doing multiplication and division, so Chris adapted the exercises inthe textbook and made worksheets of problems involving Easter


bunnies. For the group that had got up to the section in the textbook onmoney problems, Chris’ worksheets were all about buying Easter eggs.

(Ainley, 1982, p. 7)

Rich contexts: Hans Freudenthal

Freudenthal mused on how context-rich mathematics instruction becamerich contexts, perhaps influenced by the notion of rich structures. Here helists features of context-rich tasks.

Fraught with relations was the term I chose for the mathematics Iwanted to be taught. In the meantime that term has become mathematicsin rich contexts. …

[ … ]Location: a meaningful gathering of situations, which can be handled

separately, or in more or less close connection with each other. …Story: … rather than a gathering, something that, reeled off as a

succession of worksheets, is structured in time. It may be a true story orfiction, a classic or invented ad hoc. [For example, a giant leaves tracesof visiting the classroom and from his footprint his size is gauged, theamount of food he needs, etc., and then messages appear from himasking various questions.]

Project: … reality to be created [through making something].Theme: … a mathematically oriented strand of subject matter with

varying relations to reality [such as ‘light and shadow’ or ‘exponentialfunctions’].

Clippings: mainly from newspapers and weeklies, but also from booksand other media. …

Contexts were defined as domains of reality disclosed to the learner inorder to be mathematised. In the cases of location, story, project, andtheme such domains are purposefully – sometimes artificially – delimitedby the teacher or developer, who wants the learner to reinvent certainprocesses and products of mathematising. The case of clippings is a bitdifferent. Here it is not a domain but a small piece that is cut out,although its paradigmatical value for mathematising and for acquiring amathematical attitude may be enormous in comparison.

But in all cases it should be kept in mind that context is not a meregarment clothing nude mathematics, and mathematising is quite anotherthing than simply unbuttoning this garment. … Viewing context asnoise, apt to disturb the clear mathematical message, is wrong; thecontext itself is the message, and mathematics a means of decoding.



Angle as problématique: Nicolas Balacheff

The following extracts are taken from a paper in which Balacheff sketchedthe background assumptions of his approach, and applied the approach tothe case of the size of angles and the invariance of the sum of the angles of atriangle.

First, some of the background, then a proposal to make use of the learners’initial wrong conceptions by identifying four constraints.

Pupils’ conceptions of the notion of angle are likely to lead them toassert that the larger the triangle, the larger the sum of the angles … .Because of this conception, the value of a proof proposed by theteacher, even after some manipulations [learners folding triangles, etc.]are doubtful, because (a) the assertion itself might appear arbitraryinsofar as results like 182° or 178° are pragmatically as good candidatesas 180°, and (b) the pupils will be left with an open conflict betweentheir intuition … and the authority of the proposed proof.

[ … ]

1 It is not possible to tell the pupils beforehand that the purpose of thesequence will be to establish that the sum of the angles of a triangle is180°. That would destroy the problem, because the assertion wouldno longer be considered as a conjecture; the student knows theteacher always tells the truth. This is a classic example of one of thebasic beliefs held in the didactical contract.

2 The validity of the measurement of a particular set of triangles as ameans to establish the conjecture should be dismissed. But this deci-sion should be taken by the pupils on their own and not imposed bythe teacher; otherwise they will seek a proof that is acceptable to theteacher.

3 The situation we design should elicit the pupils’ conceptions aboutthe relations between the size of a triangle and the value of the sumof its angles, because it is from the contradiction between thisconception and the fact that the sums are around 180° that theconjecture could stem. This requires a situation for action.

4 We should provide the classroom with a situation for validationoriented toward the construction of a proof of the conjecture. Thatsupposes a didactical contract in which the pupils have the responsi-bility for the truth of the conjecture. This is possible only if they havehad the responsibility for forming the statement of the conjectureitself.

(Balacheff, 1990, p. 265)

Note the problem of being explicit with learners about the objectives of alesson (see also inner and outer aspects of tasks, p. 241).


Pros and cons of apparatus

Multiple embodiment: Zoltan Dienes

Dienes was led to what he called the perceptual variability principle and tothe mathematical variability principle:

… to abstract a mathematical structure effectively, one must meet it in anumber of different situations to perceive its purely structural properties.… [As] every mathematical concept involved essential variables, allthese mathematical variables need to be varied if the full generality ofthe mathematical concept is to be achieved. The application of theperceptual variability principle ensures efficient abstraction; the applica-tion of the mathematical variability principle ensures efficient general-ization. By abstraction I mean understanding the structure’s applicablebreadth; and by generalization, grasping the full extent of the mathemat-ical class, usually of numbers, for which the expression of the rule-struc-ture is valid. Effective mathematical thinking must take into account theabstraction and the generalization process.

(Dienes, 1963, pp. 158–9)

Note the use of abstraction and generalisation (see p. 59 and p. 185). Notealso the similarities with invariance as a theme in mathematics (see p. 193)and with dimensions-of-possible-variation (see p. 56). There were problemshowever, in applying and testing these principles.

Dienes reported:

We assumed throughout our experiments that abstraction would arisefrom a multiple embodiment of the concepts to be abstracted. By this Imean that situations physically equivalent to the concept-structure to belearned would, if handled according to specific instructions leadingtowards the structure, result in abstracting the common structure from allthe physical situations. We thought that when this had been accom-plished, symbolism could be introduced to describe the structure justabstracted. But as we observed the children going through such ‘abstrac-tion exercises’, it soon became clear that the picture was far morecomplex than we had assumed.

At first there was the theoretical difficulty of finding the criterion fordetermining whether a concept had been learned as an abstract structureor not. According to our first ‘simple-minded’ theory, abstraction wastested by children’s ability to transfer a mathematical structure embodiedin one situation to a different embodiment. … there are certain difficul-ties about this procedure, and, even theoretically, in order to make theprocedure valid some method of comparing embodiments would haveto be devised. Clearly some embodiments are closer to each other than


others: in one case, for example, only a small number of the attributes[relevant] to the structure may have been varied, and in another a largenumber of these attributes might have been varied. It seems that it wouldbe possible to make an embodiment so ‘noisy’ that extracting thelooked-for mathematical structure might be well-nigh impossible.

(ibid., pp. 68–9)

See also Dienes on generalization and symbols (p. 136).Not everyone agrees with the universal value of using apparatus in class-

rooms. Embodiment is a feature of the perceiver, not the apparatus itself.(See also mathematising, p. 256 and p. 283.)

Blocks abstract for some: John Holt

… Children who already understand base and place value, even if onlyintuitively, could see the connections between written numerals andthese blocks. … But children who could not do these problems withoutthe blocks didn’t have a clue about how to do them with the blocks.

… They found the blocks, … as abstract, as disconnected from reality,mysterious, arbitrary, and capricious as the numbers these blocks weresupposed to bring to life.

(Holt, 1964, pp. 218–19)

Passing over situated knowledge: Koeno Gravemeijer

Even though actions on objects, which may, of course, be mental or symbolicas well as material, lie at the heart of learning, the use of apparatus to embodymathematical concepts has been more praised than used. Dienes was not theonly person to encounter problems in researching the use of apparatus.Koeno Gravemeijer is a Dutch mathematics educator and researcher whoworked with Hans Freudenthal, and continues his work at the FreudenthalInstitute. Here he summarises his position after giving many examples:

… the use of manipulatives does not really help students attain mathemat-ical insight. … [The reason put forward is that] the mathematics embeddedin the models is not concrete for the students. … the manipulativesapproach passes over the situated, informal knowledge of the students. …

(Gravemeijer, 1994, p. 77)

Beginning with the concrete: John Dewey

Here Dewey works on the distinction between the concrete and the abstract,and challenges the notion that apparatus in itself aids learning (see alsoDewey, p. 45, and Gattegno, p. 134).


Since the concrete denotes thinking applied to activities for the sake ofdealing with difficulties that present themselves practically, ‘begin withthe concrete’ signifies that we should, at the outset of any new experi-ence in learning, make much of what is already familiar, and if possibleconnect the new topics and principles with the pursuit of an end in someactive occupation. We do not ‘follow the order of nature’ when wemultiply mere sensations or accumulate physical objects. Instruction innumber is not concrete merely because splints or beans or dots areemployed. Whenever the use and bearing of number relations areclearly perceived, a number idea is concrete even if figures alone areused. Just what sort of symbol it is best to use at the given time – whetherblocks, or lines, or figures – is entirely a matter of adjustment to a givencase. If the physical things used in teaching number or geography oranything else do not leave the mind illuminated with recognition of ameaning beyond themselves, the instruction that uses them is asabstruse as that which doles out ready-made definitions and rules, for itdistracts attention from ideas to mere physical excitations.

(Dewey, 1933, pp. 224–5)

There are interesting parallels with Mason (1980) where he suggested thatsymbols can actually be ‘concrete’, imagistic or symbolic, or all threesimultaneously.

The notion that we have only to put physical objects before the senses inorder to impress ideas upon the mind amounts almost to a superstition.The introduction of object lessons and sense-training scored a distinctadvance over the prior method of linguistic symbols, but this advancetended to blind educators to the fact that only a halfway step had beentaken. Things and sensations develop the child, indeed, but only whenhe uses them in mastering his body and coordinating his actions. Contin-uous occupations involve the use of natural materials, tools, modes ofenergy, and do it in a way that compels thinking as to how they relate toone another and to the realization of ends. But the mere isolated presen-tation of things to sense remains barren and dead.

(Dewey, 1993, p. 225)

Appropriateness?: Kath Hart

Kath Hart (b. London, 1934–) achieved the first Ph.D in mathematics educa-tion in England. She advocated the design of textbooks and tasks on thebasis of research, and led various teams to that end herself (Hart, 1981;1984). Here she challenges the use of apparatus:

Before we recommend to teachers that they use manipulatives weshould advise them to view the appropriateness and limitations of the


materials for the purpose of leading to and authenticating a part offormal mathematics.

[ … ]We need to research when manipulatives are appropriate as well as

the balance of time given to different activities within the same scheme.(Hart, 1993, pp. 27–8)

Hart’s researches showed that time spent on the apparatus was rarelymatched with time spent articulating the formula or rule. Furthermore, child-ren with difficulties in applying rules were often sent back to concrete aids,which presupposed that they remembered the connection and that theycould re-invent the it.

Need for sense-making first: Grayson Wheatley

Although the use of apparatus is promoted to counter learners’ sense ofabstractness, alienation and irrelevance, Wheatley suggested that abstractionis inescapable (see also alien, p. 102, and motivation, p. 99), and that aneffective approach promotes sense-making first and foremost.

Often the use of manipulatives is supposed to make the abstract formula-tions of mathematics comprehensible to students. Using concrete objectsto ‘show’ students a mathematical concept or relationship is still based onthe ‘abstract-first’ conception of learning. When, as Gravemeijer (1990)suggests, emphasis is placed on mathematizing from potentially mean-ingful situations, students have the opportunity to construct experience-based knowledge. … In mathematics learning, the intention to makesense is essential (Erlwanger, 1973). Neither the abstract-first nor proce-dures-first approach to learning fosters the intention to make sense. …

(Wheatley, 1992, p. 533)

Often when manipulatives are used in teaching mathematics, the teacherdemonstrates the way they are to be used and students are left littlefreedom to give meaning to the experience in ways that make sense tothem; the way the materials are to be used is prescribed. There is themistaken belief on the part of the teacher that the mathematics isapparent in the materials, for example, ‘base 10’ blocks (Cobb, Yackeland Wood, 1992). This is based on the belief that mathematics is ‘outthere’ and that models ‘show’ the concepts. The demonstration withconcrete materials is quite appealing because the concepts are so vividfor those who have already made the construction. Thus there is themistaken belief that since we, as adults, can see the mathematics in theblocks, the students will too. But the ‘seeing’ requires the very construc-tion the activity is intended to teach.

(ibid., p. 534)


Where apparatus can become an obstacle rather than an aid there are simi-larities with Fischbein’s analysis of intuition (see p. 63).

Empirical abstraction: Jean Piaget

Piaget writes about deriving information from apparatus.

It is notable … that empirical abstraction, whatever the level involved,never comes into operation by itself. In order to derive information froman object, and even if it can only be drawn from that object, the use of anassimilatory apparatus is indispensable. This assimilatory apparatus is ofa mathematical nature … a whole range of instruments … which isnecessary to the very ‘reading’ of experience itself and which is inde-pendent of other interpretations which will follow. These recordinginstruments make possible only the empirical type of abstraction, but itis clear that they themselves are not derived from the object, since theyconstitute the conditions preliminary to the subject’s cognitive grasp ofthat object. They are thus due to the subject’s own activities, and, assuch, they arise from previous reflecting abstraction. This will be true …even if the empirical abstraction which they make possible subsequentlydraws its products from the external object alone.

(Piaget, 1980, pp. 90–1)

Assimilation is a key process of Piaget (see p. 149), as is reflective abstraction(see p. 171).

Manipulatives: Patricia Moyer-Packenham

Moyer-Packenham taught in elementary school for ten years before movinginto teacher education in both mathematics and science leadership.

… Manipulatives are not, of themselves, carriers of meaning for insight.‘Although kinesthetic experience can enhance perception and thinking,understanding does not travel through the finger tips and up the arm’(Ball, 1992, p. 47). It is through their use as tools that students have theopportunity to gain insight into their experience with them. Research hasshown that for children to use concrete representation effectively withoutincreased demands on their processing capacity, they must know thematerials well enough to use them automatically (Boulton-Lewis, 1998). Ifthe user is constantly aware of the artifact then it is not a tool, for it is notserving the purpose of enabling some desired activity which moves onetoward a desired goal state (Winograd and Flores, 1986). …

Students sometimes learn to use manipulatives in a rote manner, withlittle or no learning of the mathematical concepts behind the procedures(Hiebert and Wearne, 1992) and the inability to link their actions with


manipulatives to abstract symbols (Thompson and Thompson, 1990).This is because the manipulative is simply the manufacturer’s represen-tation of a mathematical concept that may be used for different purposesin various contexts with varying degrees of ‘transparency’. …

(Moyer, 2001, pp. 176–177)

So rote learning can apply to manipulating physical objects as well as tosymbols as marks on paper. It is not just apparatus that requiresinterpretation:

Algebraic symbols do not speak for themselves. What one actually seesin them depends on the requirements of a specific problem to whichthey are applied. Not less important, it depends on what one is preparedto notice and able to perceive.

(Sfard, 1994, p. 192)

Moyer-Packenham pointed out that the teacher’s attitude and reasons forusing apparatus is likely to influence the affordances (see p. 246) available inthe situation. In a small study of pre-service teachers’ views about apparatus,she found a persistent orientation towards the ‘fun dimensions’.

In many instances teachers indicated that the use of manipulatives was‘fun’. Initially the term ‘fun’ seemed to indicate that teachers and studentsfound enjoyment in using the manipulatives during mathematics teachingand learning. Further analysis of the data suggested that embedded inteachers’ use of the word ‘fun’ were some unexamined notions that inhibitthe use of manipulatives in mathematics instruction. Teachers madesubtle distinctions between ‘real math’ and ‘fun math’, using the term ‘realmath’ to refer to lesson segments where they taught rules, procedures andalgorithms using textbooks, notebooks, worksheets, and paper-and-pencil tasks. The term ‘fun math’ was used when teachers described partsof the lesson where students were having fun with the manipulatives.

Moyer, 2001, p. 185)

See also activity theory and the mediation of tools (p. 85 and p. 220).

Advance organisers: David Ausubel

However tasks are presented, many learners find it helpful to have an overviewof what is expected before they dive into details. Ausubel, an America psycholo-gist much influenced by Piaget, called these advance organisers. They havebecome institutionalised in radio and television programmes in which you aretold what the programme is about or what someone is going to say (advanceorganiser, sign posting), then there is the substance of the programme, followedby a summary of what it was about or what the person said.


… These organizers are introduced in advance of learning itself, and arealso presented at a higher level of abstraction, generality, and inclusive-ness; and since the substantive content of a given organizer or series oforganizers is selected on the basis of their suitability for explaining, inte-grating, and interrelating the material they precede … , this strategysimultaneously satisfies the substantive as well as the programmingcriteria for enhancing the organization strength of cognitive structure.

(Ausubel, 1963, p. 81)


Affordances are possibilities due to relationship between person and situation,so tasks and texts, and particularly, apparatus, only have affordances in thecontext of particular people making use of them. Secondary play is an impor-tant part of encountering and getting to know any object, whether apparatus,images, or symbols.

Various dimensions of tasks each provide a spectrum of possibilities, alldirected towards engaging the learner in exploring, connecting and experi-encing. Apparatus (examples in the material world) may help link to learnerexperience, but has to be transcended and left behind (move into mental andsymbolic worlds) for useful learning to occur. Apparatus use can be anobstacle to learning, particularly where teachers treat apparatus as ‘for fun’rather than drawing out mathematical structure.

10 Sustaining mathematicalactivity Sustaining mathematical activity

Introduction

This chapter addresses the core of teaching. Several frameworks that haveproved to be useful in planning lessons are introduced, building on previouschapters.

The structure of any domain of knowledge may be characterized in threeways, each affecting the ability of any learner to master it: the mode ofrepresentation in which it is put, its economy, and its effective power.Mode, economy, and power vary in relation to different ages, to different‘styles’ among learners, and to different subject matters.

(Bruner, 1966, p. 44)

Integrating frameworks

Four closely interlinked frameworks are considered here. The words in eachframework have been chosen to trigger the framework into mind while plan-ning lessons and, in the midst of a lesson, to act as a remninder about whatactions could be chosen to support effective learning rather than followingautomated and habitual routines.

Enactive–Iconic–Symbolic (EIS): Jerome Bruner

The EIS framework is an expression of three worlds of experience (see p. 73).

Any domain of knowledge (or any problem within that domain ofknowledge) can be represented in three ways: by a set of actions appro-priate for achieving a certain result (enactive representation); by a set ofsummary images or graphics that stand for a concept without defining itfully (iconic representation); and by a set of symbolic or logical proposi-tions drawn from a symbolic system that is governed by rules or laws forforming and transforming propositions (symbolic representation). Thedistinction can most conveniently be made completely in terms of a

balance beam, … . A quite young child can plainly act on the basis of the‘principles’ of a balance beam, and indicates he could do so by beingable to handle himself on a see-saw. He knows that to get his side to godown farther he has to move out farther from the center. A somewhatolder child can represent the balance beam to himself either by a modelon which rings can be hung and balanced or by a drawing. The ‘image’of the balance beam can be varyingly refined, with fewer and fewer irrel-evant details present, as in the typical diagrams in an introductory text-book in physics. Finally, a balance beam can be described in ordinaryEnglish, without diagrammatic aids, or it can be even better describedmathematically by reference to Newton’s Law of Moments in inertialphysics. Needless to say, actions, pictures, and symbols vary in difficultyand utility for people of different ages, different backgrounds, differentstyles.

(Bruner, 1966, pp. 44–5)

What does it mean to translate experience into a model of the world? …there are probably three ways in which human beings accomplish thisfeat. The first is through action. We know many things for which wehave no imagery and no words, and they are very hard to teach anybodyby the use of either words or diagrams and pictures. If you tried to coachsomebody at tennis or skiing or to teach a child to ride a bike, you willhave been struck by the wordlessness and the diagrammatic impotenceof the teaching process. … There is a second system of representationthat depends upon visual or other sensory organization and upon theuse of summarizing images. … We have come to talk about the first formof representation as enactive, the second as iconic. Iconic representationis principally governed by principles of perceptual organization and by… economical transformations in perceptual organization … – tech-niques for filling in, completing, extrapolating. Enactive representationis based, it seems, upon a learning of responses and forms ofhabituation.

Finally, there is representation in words or language. Its hallmark isthat it is symbolic in nature, with certain features of symbolic systemsthat are only now coming to be understood. Symbols (words) are arbi-trary, … they are remote in reference, and they are almost always highlyproductive or generative in the sense that a language or any symbolsystem has rules for the formation and transformation of sentences thatcan turn reality over on its beam ends beyond what is possible throughactions or images. A language, for example, permits us to introducelawful syntactic transformations that make it easy and useful to approachdeclarative propositions about reality in a most striking way. We observean event and encode it – the dog bit the man. From this utterance we cantravel to a range of possible recodings – did the dog bite the man or didhe not? If he had not what would have happened? And so on. …

Sustaining mathematical activity 261

I should also mention one other property of a symbolic system – itscompactibility – a property that permits condensations [formulae] … .

(ibid., pp. 10–12)

Here Bruner is acknowledging the importance of the use of language todistance the speaker from an action (‘everything said is said by an observer’,see p. 70).

How transitions are effected – from enactive presentation to iconic, andfrom both of these to symbolic – is a moot and troubled question. To putthe matter very briefly, it would seem as if some sort of image formationor schema formation – whatever we should call the device that renders asequence of actions simultaneous, renders it into an immediate represen-tation – comes rather automatically as an accompaniment of response tostabilization. But how the nervous system converts a sequence ofresponses into an image or schema is simply not understood.

(ibid., p. 14)

Bruner was pointing to the fundamental problem of how it is that a sequenceof impressions becomes a self-contained entity with stressing and ignoringproducing a sense of sameness and of difference in relation to past experi-ence (see reification, p. 167, and discernment, p. 55). Bruner advocatedpaying attention to transitions between enactive participation (learnersdoing things like moving or counting bricks, even doing calculations in thehead or on a calculator) and iconic participation, imagining without actuallydoing. The word ‘transition’ should not be taken as describing a once and forall shift between worlds, but rather a gradual shift as to which world is domi-nant at the time. A similar ‘transition’ takes place between the iconic and thesymbolic as the ‘doing’ is recorded in general terms, formalised in some way.

The following three frameworks were developed for an Open Universitycourse for teachers in the 1980s to support teachers in the developmentswhich Bruner suggests are problematic.

Do–Talk–Record (DTR)

The Do–Talk–Record framework (and variants such as See–Say–Record) wasproposed as useful for remembering to get learners talking about their ideasbefore rushing into symbols and written records, and for justifying spendingtime in this way. The idea is expressed as:

Action and concrete experience of a process, linked through language devel-opment and much discussion, support pictures of that action and wordswhich describe the process, which in turn is linked as a result of frequent useand successive shorthanding to standard notation and layout of an algorithm.

(based on Floyd et al., 1982, Block 1, p. 23)


Note the similarity with other versions of reification (see p. 167).

Students on EM235 were led to a framework of Do–Talk–Record in orderto draw attention to the desirability of learners acting upon familiarobjects and talking about what they are doing in order to develop andintegrate the basic language patterns of the topic into their own func-tioning. Recording is then motivated by giving instructions to others asto how and when to perform the same actions using longhand storiesand pictures, and then, through successive shorthanding, to approachstandard notation and layout.


New ideas take time to grasp. Useful activities are ones which provideplenty of opportunity for rehearsing language, rather than filling inblanks in a worksheet. For example, games in which each learner cancheck that others are using the language appropriately, providing varia-tions of activities in order to keep learners in a situation long enough forthem to become fluent in the use of the language.


See–Experience–Master (SEM)

No idea is grasped or appreciated on first or even second encounter. TheSEM framework, also used on The Open University Course EM235, can serveas a reminder not to expect too much of learners too quickly.

First encounters with any idea are at best transitory. A word you do notrecognize flashes by in a conversation; a concept you vaguely recognizegets used several times, but you have no clear idea of its full import; a task isset and although you can work out what to do you are not at all clear whatit is really about or why it works, or even, perhaps, what it achieves. It isunreasonable then to expect learners to grasp ideas on first encounter.

However, as you get more experience with a concept or a technique,it gradually starts to fit in with what you already know. You start toassimilate it into your familiar ways of thinking, or you find yourselfadjusting your thinking to accommodate the new idea. … Familiarityand facility arise only after considerable experience and, usually, reflec-tion on that experience. For one thing we do not seem to learn fromexperience is that we do not often learn from experience alone. Some-thing else is required.

Sometimes you find that after a period of time, what seemed awkwardand a struggle suddenly comes easily. Learning can be thought of as amaturation process, like the baking of bread and the brewing of beer:some transformation is required which cannot be rushed.

(based on Floyd et al., 1982, Blocks 2 & 4)


Transitions between the three modes of representations in the threeworlds introduced by Bruner can be enhanced and supported by payingattention to the purpose of ‘doing’, of manipulating confidence-inspiringentities, namely, to get a sense of what the particular is indicating might betrue in general, and eventually articulating this. A new concept is appreciatedby reference to familiar examples and how the examples illustrate theconcept; a difficult problem is tackled by simplifying and particularising it soas to get a sense of what might be going on, before returning to the moredifficult or more general case; a tricky technique is mastered by applying it tofamiliar cases and then to less and less familiar cases as a sense of what thetechnique does and how it works and is used, develops, eventually comingto articulation (you express it for yourself).

Manipulating–Getting-a-sense-of–Articulating (MGA) spiral

This next framework was also developed by Floyd et al. (1982) and elabo-rated in various places, though the ideas have ancient roots in the wisdom ofteachers through the ages. Its role is to serve as a reminder of the purpose of‘doing things’ beyond the mere accomplishment of the task, and to offer a meta-phor for the complex layering process which we call experience and learning.

Picture mathematical thinking on a helix which loops round and round.Each loop represents an opportunity to extend understanding byencountering an idea, an object, a diagram, or symbol with enoughsurprise or curiosity to impel exploration of it by manipulating. Thelevel at which manipulation begins must be concrete and confidenceinspiring and the results of the manipulation will then be available forinterpretation. Tension provoked by the gap which opens between whatis expected and what actually happens provides a force to keep theprocess going and some sense of pattern or connectedness releases thetension into achievement, wonder, pleasure, further surprise or curiositywhich drives the process on. While the sense of what is happeningremains vague, more specialization is required until the force of thesense is expressed in the articulation of a generalization. Articulations donot have to be verbal. They might well be concrete, diagrammatic orsymbolic but they will crystallize whenever is the essence underlying thesense which has been achieved as a result of the manipulations. Andachieved articulation immediately becomes available for new manipu-lating, and the wrap-around of the helix. Each successive loop assumesthat the thinker is operating at a deeper level of complexity. Theconnectedness of the loops always permits the thinker the opportunityto track back to previous levels and therefore to revise articulating thatmight have begun to wobble.


(Mason, Burton and Stacey, 1982, pp. 155–6)

Interconnections

Bruner’s three modes of representation (EIS) allied to the three worlds (seep. 73) are embedded in what the MGA spiral offers:

Turning to enactive elements to explore the meaning of symbols orconcepts; Using enactive elements to try to get a sense of pattern; Askingfor images, metaphors, diagrams to illustrate what is going on;crystallizing understanding in symbolic form; practising with examplesto move the symbolic form into enactive elements.

(Mason, 1980, p. 11)

These in turn are related to Do–Talk–Record (see p. 262) as a summary ofconjecturing (see p. 139):

1 Do examples (Specialize) using entities with which you are entirelyconfident, which you can manipuilate easily while part of your atten-tion remains focused on your primary goal.

2 Try to get a sense of underlying pattern or relationship. Oftendiagrams or metaphors will help here – have you seen a similarproblem or idea? An analogous one?


3 Try to articulate the pattern you sense. Keep refining the articulationuntil it can be checked on examples [taking you] back to 1 again.

(ibid., p. 10)

The first is mostly doing (but with attention: recall the two birds, p. 32) affordingopportunity for seeing (encountering) new ideas, experiencing themes andideas met previously (as well as contexts and ways of speaking and so on – seeStructure-of-a-topic, p. 203), and gaining mastery of techniques already encoun-tered. The second is enhanced by talking (to yourself and to others) in order toget a sense of the pattern or multiple relationships, to experience multipledimensions-of-possible-variation (see p. 56) which make up or contribute to theconcept involved. It supports educating awareness (see p. 61 and p. 204). Thethird, articulating, leads to recording and formalisation, through recognition ofproperties independent of the particular objects used, and hence to apprecia-tion of generality. Note connections with reification (see p. 167).

For a summary and synthesis of the three frameworks, see Floyd et al.(1982, Block 4, pp. 20–27).

Teacher intervention

Scaffolding

The heart of teaching lies in teacher interventions. But how to describe theseinterventions let alone recommend some over others is highly problematic.

The hardest part of teaching by challenging is to keep your mouth shut,to hold back. Don’t say, ask! Don’t replace the wrong A by the right B,but ask, ‘Where did A come from?’ Keep asking ‘Is that right?, Are yousure?’ Don’t say ‘No’; ask ‘Why?’

(Halmos, 1985, p. 272)

Scaffolding: David Wood, Jerome Bruner and Gail Ross

Wood, Bruner and Ross introduced the term scaffolding into educationalliterature. The term has been used in a wide range of senses, though


The three forms of representation, Enactive–Iconic–Symbol, correspond to thethree worlds. To fully encounter an idea learners need to engage in Doing–Talking–Recording, but it is important not to rush to recording. See–Experi-ence–Master is a reminder not to expect too much from early encounters:teaching takes place in time while learning takes place over time.

Manipulate–Get-a-sense-of–Articulate, Do–Talk–Record, Enactive–Iconic–Symbolicand See–Experience–Master together provide a rich collection of triggers to sensitiseawareness of opportunities when planning and in the midst of lessons.

originally it was intended to describe how a teacher works in relation toVygotsky’s notion of the zone of proximal development (see p. 88).

Well-executed scaffolding begins by luring the child into actions thatproduce recognizable-for-him solutions. Once that is achieved, the tutorcan interpret discrepancies to the child. Finally, the tutor stands in aconfirmatory role until the tutee is checked out to fly on his own … .

(Wood, Bruner and Ross, 1976, p. 96)

1 Recruitment. The tutor’s first and obvious task is to enlist the problemsolver’s interest in and adherence to the requirements of that task. …

2 Reduction in degrees of freedom. This involves simplifying the task byreducing the number of constituent acts required to reach solution. …

3 Direction maintenance. Learners lag and regress to other aims, givenlimits in their interests and capacities. The tutor has the role ofkeeping them in pursuit of a particular objective. Partly it involveskeeping the child ‘in the field’ and partly a deployment of zest andsympathy to keep him motivated. …

(ibid., p. 98)

This is part of being ‘consciousness for two’ (see p. 88).

4 Marking critical features. A tutor by a variety of means marks oraccentuates certain features of the task that are relevant. His markingprovides information about the discrepancy between what the childhas produced and what he would recognize as a correct production.His task is to interpret discrepancies.

(ibid., p. 98)

Note parallels with stressing and ignoring (see p. 127) and structure of atten-tion (see p. 60).

5 Frustration control. There should be some such maxim as ‘Problemsolving should be less dangerous or stressful with the tutor than with-out’. Whether this is accomplished by ‘face saving’ for errors or byexploiting the learner’s ‘wish to please’ or by other means, is of onlyminor importance. The major risk is in creating too much depend-ency on the tutor.

6 Demonstration. Demonstrating or ‘modeling’ solutions to a task,when closely observed, involves considerably more than simplyperforming in the presence of the tutee. It often involves an ‘idealiza-tion’ of the act to be performed and it may involve completion oreven explication of a solution already partially executed by the tuteehimself. In this sense, the tutor is ‘imitating’ in idealized form anattempted solution tried (or assumed to be tried) by the tutee in the


expectation that the learner will then ‘imitate’ it back in a moreappropriate form.

(ibid., p. 98)

Scaffolding: David Wood and David Middleton

David Wood has devoted his entire career to exploring aspects of tutoring asit happens between mothers and young children, and between teachers andyoung children. However much of what is observed applies to tutoring at allages.

The child, of course, is no passive observer … . The change and develop-ment of his abilities are not to be viewed as ‘copying’ or imitation in thesense that he merely matches his behaviour to that of another. Rather, …he emerges as an active, selective agent, more akin to a ‘rule-inducer’ thana mere copier of action patterns (Wood and Middleton, 1975). However, itis our contention that any process of rule induction or problem-solving onhis part can, and indeed often must, be facilitated by the intervention ofanother who is more expert in the situation than he is. It is the aim of thepresent study to examine in some detail the strategies which the mothersdisplay when they attempt to fulfil this instructional role.

[ … ]Clearly, when a child is alone success demands that he perform …

operations himself. But when a mother intervenes to help him she maytake over one or more of them. By so doing, she may leave the childrelatively free to concentrate all his attention and effort upon a narrowerrange of alternatives within the task. Where, for example, she asks himin general terms for some activity, ‘Can you make some more like thisone?’, she merely suggests a relatively short-term goal to the child,leaving to him the task of determining the parameters for search,searching, assembling and evaluating. Or she might ask for somethingmore specific like a ‘big one’, further constraining his actions by deter-mining not only a goal but at the least some of its attributes.

(Wood and Middleton, 1975, pp. 181–2)

This is part of being ‘consciousness-for-two’ (see p. 88). This leads to a coreproblem in teaching: when and how to intervene?

The crucial problem facing an instructor, of course, is deciding at whatlevel to intervene. How many operations should she ask of the child andhow many should she do herself? On the one hand, she should not wantto see him ‘swamped’ by too many possibilities, but neither too shouldshe wish to stifle his performance by doing too much herself. … Ideally,the instructor should engineer discrepancies for the child by constantlyshowing or requesting goals which he can currently recognize but not


produce. In this way, he can lead the child to suitably constrainedproblem-solving activity.

(ibid. p. 182x)

Note the recourse to disturbance (see p. 55, p. 101 and p. 161). Howeverdesirable, ‘doing for learners only what they cannot yet do for themselves’ isnot an easy path to follow.

… we suggest that, although any effective instruction requires the childto do more than he is immediately capable of, it must not ask him for toomuch. Ideally, the child should be asked to add one extra operation ordecision to those which he is presently performing. This level of inter-vention we have termed the ‘region of sensitivity to instruction’ and ourhypothesis is that the most effective instructors will concentrate theirinstructional activity within this region.

(ibid., p. 182)

The ‘region of sensitivity to instruction’ is informed by the notion of near-simultaneity of variation within dimensions-of-possible-variation (see p. 56).The purpose of scaffolding is to provide temporary support, which musttherefore fade as the learner gradually takes over. Otherwise the learner istrained in dependency rather than gaining independence.

Questioning questioning

One obvious form of scaffolding (see p. 266) is asking questions. Askinglearners questions is entirely natural, even if not always productive or effective.

Moving beyond the particular: Jerome Bruner

… if you … are convinced that the best learning occurs when the teacherhelps lead the pupil to discover generalizations on her own, you’relikely to run into an established cultural belief that the teacher is anauthority who is supposed to tell the child what the general case is,while the child should be occupying herself with memorizing the partic-ulars. And if you study how most classrooms are conducted, you willoften find that most of the teacher’s questions to pupils are about partic-ulars that can be answered in a few words or even by ‘yes’ or ‘no’. Soyour introduction of an innovation in teaching will necessarily involvechanging the folk psychological and folk pedagogical theories ofteachers – and, to a surprising extent, of pupils as well.

(Bruner, 1996, p. 46)


Questioning: Janet Ainley

Question asking turns out to be very complicated, largely because whatappears to be a question can be an instruction, and what appears to be aninstruction can be a question. Furthermore the purpose of asking questionscan be very varied. Here Ainley, who worked with Skemp, analyses thenature and purposes of questions.

… The students know that the teacher already knows the answer to thequestions she is asking. Asking questions to which you already knowthe answer is a very odd linguistic activity, almost entirely restricted toclassrooms, or at least to teaching situations. In other circumstances itwould rightly be considered bizarre, except as a conversational gambit(where it is not apparent to the person you are talking to that you doknow the answer). And yet this activity is what is generally meant by‘questioning’ children. It is part of the ‘school game’ and teachers andstudents both know its purpose: the teacher does not want to find outinformation, but rather to ascertain whether or not the students knowthe answers. …

[ … ]… what constitutes an open rather than a closed question, since being

open or closed seems not to be so much an attribute of the question per seas something read into it by the questioner or the hearer. Teacher andpupils may have different perceptions of this distinction, and students’perception may be altered by the way in which their answers are handled.The purpose of these questions is not to gain new information. This iswell known to the students, who may very well perceive such a questionas a closed one, that is, as having only one correct answer. When startingoff investigations … I often ask, ‘How many different ones can you find?’My intention is to open up the investigation, but some hearers will takethis as a closed question, requiring an exact numerical answer. …

… There is an important sense in which any type of questioningconveys information. By asking questions you indicate what is ofinterest to you. When the teacher asks a question, she is drawing atten-tion to those aspects of the situation which are important. This is trueeven with more open questions. A teacher’s question conveys themessage ‘there’s something important here, and you should know aboutit’. Again, this is emphasised by the fact that the student believes theteacher already knows the answer(s). One reason for asking such aquestion is precisely to draw attention to something.

[ … ]… There seem to be, in very general terms, three distinct types of

activities which have the same syntactic form, but serve differentpurposes. These we might call,


• genuine questions – asked because you want to know the answer;• test questions – asked when you already know the answer, to find out

if the other person knows it;• provoking questions – asked to draw attention to something you want

the person to think about.(Ainley, 1987, pp. 25–6)

Open and closed questions: Anne Watson

As we have seen, questions are often described as either open or closed, butit is not at all clear that open questions are more effective in involvinglearners than are closed questions, or vice versa.

An open question is usually taken to mean one with several answers, towhich many learners can contribute, but contrast these two open questions:

If the answer is 4, what could the question be?

I want you to make up three questions to which the answer is 4, and eachquestion must come from a different topic we have studied this term.

The first question is wide open, and is likely to generate low levelarithmetical operations using small whole numbers. The second is moreconstrained and denies the possibility of sticking with simple operations;learners are forced to think beyond the obvious. Both are open, with theadvantages of open questions, but one is more likely to involve grap-pling with concepts than the other.

Compare the first open question to this closed question:

What number is a square number, and is also the number of sides of ashape which can be made by sticking two congruent triangles togetheredge-to-edge?

The latter question is closed, but encourages engagement withconcepts. The almost Orwellian mantra ‘open-good; closed-bad’ isclearly misleading.

(Watson, 2002, pp. 34–5)

Learners who are used to directed instructions may at first be flummoxed byquestions which invite them to make choices. But over a period of timewhen they find that they are permitted to make choices, their creativity andadventurousness is likely to open out.

Any question can be closed down so that a single yes or no is all that isrequired, and learners get quite good at guessing which it is likely to be by thetone of voice of the questioner, pauses and emphasis and so on. It is equallythe case that any question or instruction, no matter how narrow and restrictive,can be opened out. One obvious device is to ask learners to answer the


question in as many different ways as possible. Another is to ask them toconsider various dimensions-of-possible-variation and to make up their ownquestions to illustrate these different dimensions (see p. 56).

Sources of questions: John Mason

Questions are often asked because of some internal prompt:

Conjecture: an adult asks a learner a question when the adult, in thecompany of the learner, experiences a shift in the focus of their ownattention. The question is intended to reproduce that shift of focus in thelearner.

In particular, enquiry-questions are asked when people becomeaware that they are uncertain, confused, stuck, struck by something theycannot account for or realise that some expectation is contradicted.

(Mason, 2002b, p. 248)

Learners’ habits: John Holt

Children can develop deeply ingrained habits that serve them well in mostclassrooms. There may be circumstances in which these habits are nothelpful, but it takes a determined teacher to break out of the potentialstraightjacket.

John Holt describes his interactions with a learner, revealing some of thehabits to which the learner resorted:

This child must be right. She cannot bear to be wrong, or even toimagine that she might be wrong. When she is wrong, as she often is, theonly thing to do is to forget it as quickly as possible. Naturally she willnot tell herself that she is wrong; it is bad enough when others tell her.When she is told to do something, she does it quickly and fearfully,hands it to some higher authority, and awaits the magic words, ‘right’, or‘wrong’. If the word is ‘right’, she does not have to think about theproblem any more; if the word is ‘wrong’, she does not want to, cannotbring herself to think about it.

This fear leads her to other strategies, which other children use aswell. She knows that in a recitation period, the teacher’s attention isdivided among twenty students. She also knows the teacher’s strategy ofasking questions of students who seem confused, or not paying atten-tion. She therefore feels safe waving her hand in the air, as if she werebursting to tell the answer, whether she really knows it or not. This is hersafe way of telling me that she, at least, knows all about whatever isgoing on in class. When someone else answers correctly she nods herhead in emphatic agreement. Sometimes she even adds a comment,though her expression and tone of voice show that she feels this is risky.


It is also interesting to note that she does not raise her hand unless thereare at least half a dozen other hands up. …

[ … ]… A teacher who asks a question is tuned to the right answer, ready to

hear it, eager to hear it, since it will tell him that his teaching is good andthat he can go on to the next topic. He will assume that anything thatsounds close to the right answer is meant to be the right answer. So, for astudent who is not sure of the answer, a mumble may be his best bet.

(Holt, 1964, pp. 12–13)

One of the purposes of questioning is to draw learner attention to mistakes,and to discover whether they are slips or indicators of confusion or inappro-priate generalisation and reconstruction. See mistakes, p. 208 and p. 303, forhow to learn from learner mistakes.

Questioning: Amy Martino and Carolyn Maher

Carolyn Maher (b. New Jersey, 1941–) has been researching details of class-room interactions for many years. Here, writing with a colleague, shesummarises some of her research about questioning.

… Over the course of years, we have observed that a very special combi-nation of student, teacher, task, and environment fosters individualcognitive growth in the mathematics classroom. To begin, students needa classroom environment that allows them time for exploration andreinvention. The teacher in this type of class embraces the ideal thatstudents must express their current thinking. This thinking is then care-fully considered by both teacher and peers. How is this done? It beginswith teacher modeling, and very gradually this careful listening andexchange of ideas becomes the accepted mode of communication usedby all members of this community. Students begin to realize thatdiscussing their ideas and concerns with the community aids them inadvancing their own thinking. In order for this part of the learningprocess to occur, students must be willing to engage with new ideas,build models, listen to input from other sources, and sometimes exposetheir own confusions or misconceptions. This means that the student inthis learning environment is constantly rejecting, incorporating, or with-holding judgement on new ideas that arise from discussion. Naturally,this is all based upon the premise that the student cares about under-standing what he/she is studying.

(Martino and Maher, 1999, pp. 53–4)

The paper has numerous examples from the research. There are similaritieswith conjecturing (see p. 139), and making use of listening (p. 225).


Funnelling: John Holt

Inspecting his own teaching closely, Holt discovered that learners knowthat not every question has to be answered. Some learn to play the teacher-questioning game all too well: For example,

They are very good at … getting other people to do their tasks for them.I remember the day not long ago when Ruth opened my eyes. We hadbeen doing math, and I was pleased with myself because, instead oftelling her answers and showing her how to do problems, I was ‘makingher think’ by asking her questions. It was slow work. Question afterquestion met only silence. She said nothing, did nothing, just sat andlooked at me through those glasses, and waited. Each time, I had tothink of a question easier and more pointed than the last, until I finallyfound one so easy that she would feel safe in answering it. So we inchedour way along until suddenly, looking at her as I waited for an answer toa question, I saw with a start that she was not at all puzzled by what Ihad asked her. In fact, she was not even thinking about it. She wascoolly appraising me, weighing my patience, waiting for the next, sure-to-be-easier question. I thought, ‘I’ve been had!’ The girl had learnedhow to make me do her work for her, just as she had learned to make allher previous teachers do the same thing. If I wouldn’t tell her theanswers, very well, she would just let me question her right up to them.

(Holt, 1964, p. 24)

What happened to John Holt happens to every teacher at some time. Hein-rich Bauersfeld called this phenomenon funnelling.

Funnelling: Heinrich Bauersfeld

• The teacher recognizes a student with difficulties; …• The teacher opens with a short question in order to stimulate self-

correction. He receives an unsatisfactory reaction.• The teacher then goes further back to collect and clear prerequisites for

the insight, aiming at an ‘adequate’ reaction from the student. Adequateat this stage is already an approximate fit with the teacher’s expectation.

• Continued deviant answering on the student’s side meets on the teacher’sside a growing concentration on the stimulation of the ‘adequate’ answerthrough more precise, that is, narrower, questions. Thus the standard for‘adequateness’ deteriorates, the quality of the discussion decreases.

• Step by step the teacher, in fact, through what he does, reduces hispresumption of the student’s actual abilities and self-government in away that is quite the opposite to his intentions and in contradictioneven to his subjective perception of his own action (he sees himself‘providing for individual guidance’).


• The student realizes both the simplified but stiffer demands and thegrowing tension … [which] intensifies with teacher and student.

• When the deterioration has come down to the simplest exacting reci-tation or completion by the student, the culmination is reached. Justone expected word from the student can bring the teacher to apresentation of the complete solution himself.


Bauersfeld noted that the longer such a pattern continues, the less likely it is tobreak off before reaching the end where the teacher essentially gives away theanswer that has been sought all along. (See also Bauersfeld, 1980; Wood, 1998.)

Effect on thinkers: John Holt

John Holt concludes that learners are not always in class with the intention oflearning (see purposes, p. 42).

Schools and teachers seem generally to be as blind to children’s strate-gies as I was. Otherwise, they would teach their courses and assign theirtasks so that students who really thought about the meaning of thesubject would have the best chance of succeeding, while those whotried to do the tasks by illegitimate means, without thinking or under-standing, would be foiled. But the reverse seems to be the case. Schoolsgive every encouragement to producers, the kids whose idea is to get‘right answers’ by any and all means. In a system that runs on ‘rightanswers’, they can hardly help it. And these schools are often verydiscouraging places for thinkers.

(Holt, 1964, p. 25)

A useful overview of various language games like funnelling can be found inBauersfeld (1995).

Topaze effect: Guy Brousseau

Guy Brousseau described funnelling by reference to a character in MarcelPagnol’s famous French play in which the character Topaze is giving a dicta-tion test to a learner. For difficult words he pronounces the words letter byletter, thereby permitting the learner to spell correctly without effort.

The answer that the student must give is determined in advance; theteacher chooses questions to which this answer can be given. Of course,the knowledge necessary to produce these answers changes, as does itsmeaning. … If the target knowledge disappears completely, we have theTopaze effect.

(Brousseau, 1997, p. 25)


Exerting control: Philip Jackson

Philip Jackson is an experienced and sensitive observer and recorder of inci-dents in classrooms. Here is just a brief extract analysing several incidents(not included here) which show how sensitive questioning has an importantrole in socialising learners to become aware of themselves and their actions.‘Visiting’ means going to someone else’s desk and chatting.

… the children being questioned are invited by the teacher to stepoutside their own skin, to see their actions from an external perspectiveand often to give them a name or label from that perspective. The invita-tion may take the form of a fairly neutral query, as when the children aresimply asked, ‘What are you doing?’ or it may provide them with optionsto use in their description, as when Mrs Martin asks, ‘Are you visiting orhelping?’ Occasionally Mrs Martin offers her own perspective, whichreveals to the children how their actions are actually seen by someoneelse, as when she says, ‘I thought you were (looking for something)’.

Being questioned in this way encourages the children to becomejudges of their own actions. Yet the freedom to make those judgments, asthe process also makes clear, is by no means unconstrained. The catego-ries by which to judge are often set in advance and are usually few innumber. There are other people present who are doing the judging aswell – the teacher, one’s classmates who are looking on, and sometimesan adult observer or two – which means that one’s own judgment may notonly be tested against those of others but may sometimes be contested,called into question, disagreed with. The public nature of the process, thefact that the children are asked not only to judge, but also to announcetheir judgments in a voice that all can hear, not only opens the door tofalsifying one’s response, it also entails an act of commitment, a form ofgiving one’s word. In short, it calls upon the children to see themselves, ifnot as others see them, then at least as they choose in that particularcircumstance to be seen by others.

(Jackson, 1992, p. 52)

Other good sources of classroom incidents include Holt (1964; 1967; 1970)and Armstrong (1980).


Scaffolding and fading, dependence and independence are of central concernin intervention. Acting as ‘consciousness for two’ requires skilful and sensitiveintervention.

Learners already expect there to be a predetermined answer to question,which can be genuine, testing or provoking. Funnelling is an easy trap to fallinto, especially when trying to ‘make it easy’ for learners, driven by a strongdesire to make sure that they ‘understand’.

Mathematical discussion

Developing practices that were inspired by contact with Vygotsky’s ideas,Bruner and his colleagues noticed that one role of a teacher when interactingwith learners is to act as a consciousness for two (see p. 88). By resistingdoing for the learners what they can already do for themselves, the teachercan hold awareness of goals while the learner pursues subgoals.

Discussion: Janette Warden

Generating considered and considerate discussion in the classroom takestime and attention:

Before the children will give each other their full attention whenengaged in a group discussion I find that I have to get across that everymember of the class is entitled to my full attention when they arereading to me; being helped by me; or simply talking to me. On anumber of occasions, with a new class, whilst, say, hearing a child readanother child has come to ask a question, or, for help, and interrupts, Istop and tell the child not to interrupt but to wait. Afterwards I explainthat it is not easy to listen to two people at the same time, and, that ifwe are listening properly we want to give our full attention to what isbeing said to us.

During these ‘settling in’ days, whilst I am still working with the classas a group or with individual children, in order to help the children gainconfidence in their own judgments and ideas I constantly ask questionssuch as ‘Well what do you think?’ or ‘Can you explain to me how you didthis?’ Later on when group work is more established I still ask thesequestions. They really help them to start thinking for themselves, oftenfor the first time!

(Warden, 1981, pp. 249–50)

Scientific debate: Marc Legrand

Marc Legrand (b. France, 1943–) was uninterested in mathematics until histwenties when, under the influence of a particular teacher, he himselfengaged in scientific debate. With teacher and researcher colleagues he hasdeveloped techniques for engaging undergraduates explicitly in scientificdebate which he justifies as an epistemological principle:

A person who has not had sufficient occasion actually to play, in its fullgrandeur, a genuine game of science has very little chance of interestinghimself in the essential reasoning process of science, of understandingwhat the results of this reasoning really have to offer (understanding thepower, but also the limitations, of its algorithms and modes of thought),


and consequently of finding relevant ways of using those results to find ascientific solution to his own problems.

(Legrand, 1995, quoted in translation in Warfield, webref)

Scientific debate means an interaction in which conjectures are formulated,proposed, challenged, tested and justified (see conjecturing atmosphere, p.141). Learners are seen as participants in a scientific community whosemethods of development include conjectures and modifications and proofsand refutations (Lakatos, 1976). Scientific debates can arise spontaneouslywhen learners query a statement. Scientific debates can also be provokedintentionally by asking learners to make a conjecture regarding some prob-lematic issue.

The essence of a scientific debate and of a conjecturing atmosphere is thatpeople are eager to try ideas out and are neither embarrassed nor ashamedto make a mistake: everything said is offered as a conjecture, with the inten-tion of modifying it if necessary (in contrast to an ethos in which things areonly said when the sayer is confident they are correct). Those who are uncer-tain about some detail often choose to speak, while those who are confidentoften choose to listen and then to suggest modifications through counter-examples, images, questions and suggestions. The intention is to create theconditions under which fruitful mathematics is done at any level ofsophistication.

It takes time to accustom learners to mathematical discussion, but it is anessential part of teacher–learner–content interaction (see p. 221) and itsupports and makes use of learners powers to conjecture (see p. 139) and tojustify.

Scientific debate: Derek Holton

Derek Holton (b. Buckinghamshire, 1941–) is now a New Zealand mathema-tician and prolific author of pamphlets aimed at helping aspiring youngmathematicians engage in mathematical thinking and also to prepare them-selves to participate in the annual international mathematical Olympiads.Here he summarises the notion of scientific debate.

Under le débat scientifique, (see Legrand, 1993) students are seen asparticipants in a scientific community whose methods of developmentinclude conjectures, proofs and regulations. Scientific debates can arisespontaneously, as when a student asks a question, or can be intention-ally provoked. The guiding principles for scientific debate include:

• disturbance: students must encounter and deal with conflict;• inclusiveness: everyone should have an opportunity to understand

what we try to teach; and


• collectivity: collective resolution of issues shows how to work withcontradictions and to respect the views of others.

Now it may seem strange that what is labelled ‘scientific’ has such astrong social underpinning. Maybe this can be explained by noting thatthe point of the exercise is to allow students to engage in ‘scientificdebate’. This requires an atmosphere where conjecturing is supported,where students feel free to put forward their ideas, where they are notembarrassed to make a mistake, and where they feel that they are able tomodify the ideas of others. In order to generate an atmosphere in whichvaluable debate may take place, students and staff must value certainsocial principles such as respect for each other’s views.

(Holton, webref)

Note the role of disturbance (see p. 55, p. 101 and p. 161) and collective orsocial involvement in a community of practice (see p. 95).


Scientific debate is a form of conjecturing atmosphere within which mathematicalthinking thrives and develops.

11 Concluding mathematicalactivity

IntroductionConcluding mathematical activity

Mathematical activity usually concludes with reflection.Reflection is advocated by very many authors. For example, Polya (1957)

proposed four phases of problem solving: understanding the problem,making a plan, carrying out the plan, and looking back (reflection): simpleadvice, but not so easy to carry through. As many have observed, reflectionis ‘more honoured in the breach than the observance’.

Mason, Burton and Stacey (1982) expanded Polya’s four phases to sevenin order to try to make the identification of different phases more useful:getting started, getting involved, mulling, keeping going, insight, beingsceptical, and contemplating.

In the Project for Enhancing Effective Learning (PEEL) in Australia (North-field and Baird, 1992), learners were encouraged to keep diaries and toreflect on their activities; the difficult part in this project was to sustain overtreflection as a classroom practice (see also Waywood, 1992, 1994).

Li (1999, p. 33) observed from his Chinese perspective that reflection is onlyuseful if there is something specific on which to reflect. He considered thatmanipulative facility and competence are prerequisites for effective reflection.

Reflection

Reflection: John Dewey

… reflective thinking, in distinction from other operations to which weapply the name of thought, involves (1) a state of doubt, hesitation,perplexity, mental difficulty, in which thinking originates, and (2) an actof searching, hunting, inquiring, to find material that will resolve thedoubt, settle and dispose of the perplexity.

(Dewey, 1933, p. 12)

… Thinking begins in what may fairly enough be called a forked-roadsituation, a situation that is ambiguous, that presents a dilemma, that

proposes alternatives. As long as our activity glides smoothly along fromone thing to another, or as long as we permit our imagination to enter-tain fantasies at pleasure, there is no call for reflection. Difficulty orobstruction in the way of reaching a belief brings us, however, to apause. In the suspense of uncertainty, we metaphorically climb a tree;we try to find some standpoint from which we may survey additionalfacts and, getting a more commanding view of the situation, decide howthe facts stand related to one another.

(ibid., p. 14)

There are similarities with disturbance (see p. 55, p. 101 and p. 161).

… General appeals to a child (or to a grown-up) to think, irrespective ofthe existence in his own experience of some difficulty that troubles himand disturbs his equilibrium, are as futile as advice to lift himself by hisboot-straps.

(ibid., p. 15)

Dewey distinguished between reaction to a situation, and thoughtful reflec-tive response:

There may, however, be a state of perplexity and also previous experi-ence out of which suggestions emerge, and yet thinking need not bereflective. For the person may not be sufficiently critical about the ideasthat occur to him. He may jump at a conclusion without weighing thegrounds on which it rests; he may forego or unduly shorten the act ofhunting, inquiring; he may take the first ‘answer’, or solution, that comesto him because of mental sloth, torpor, impatience to get somethingsettled. One can think reflectively only when one is willing to enduresuspense and to undergo the trouble of searching. … It is at the pointwhere examination and test enter into an investigation that the differ-ence between reflective thought and bad thinking comes in. To be genu-inely thoughtful, we must be willing to sustain and protract that state ofdoubt which is the stimulus to thorough inquiry, so as not to accept anidea or make a positive assertion of a belief until justifying reasons havebeen found.

(ibid., p. 16)

Reflection: Richard Skemp

Richard Skemp was led from considering reflection into territory similar tothat of Gattegno and Bruner in relation to language as a means to distancethe speaker–thinker–reflector from the action. This echoes Maturana’s notionthat everything said is said by an observer (see p. 70). Skemp proposed thatreflection is enhanced when labels are used.

Concluding mathematical activity 281

[Reflective activity] involves becoming aware of one’s own concepts andschemas, perceiving their relationships and structure; and manipulatingthese in various ways. … the intervening processes are cognitive, and makepossible the overall activity which we call reflective intelligence. …

The process of becoming aware of one’s concepts for the first timeseems to be quite a difficult one. … the overall development of thisability extends over a number of the years of childhood. But even inpersons with highly developed reflective ability, it is still a struggle tomake newly formed, or forming, ideas conscious.

… It is largely by the use of symbols that we achieve voluntary controlover our thoughts.

Verbal thinking (which can be extended to include algebraic and anyother pronounceable symbols) is internalized speech; as may beconfirmed by watching the transitional stages in children. The use ofpronounceable symbols for thinking is closely related to communica-tion; one might describe it as communication with oneself. So becomingconscious of one’s thoughts seems to be a short-circuiting of the processof hearing oneself tell them to someone else. This view is supported by acommon observation that actually doing so to a patient listener (thinkingaloud) is nearly always helpful when one is working on a problem.Visual thinking is a much more individual matter …

(Skemp, 1971, pp. 82–3)

Reflection: Hans Freudenthal

Freudenthal suggested that reflection is going on all the time:

… when I use the word ‘reflection’, I mean mirroring oneself in someoneelse in order to look through his skin, to explore him, to take him in.And, consequently, since somebody else is like oneself – a human – thisis an experience about human behaviour and, finally, knowledge aboutone’s own behaviour. So from mirroring oneself in someone else follows– as the night the day – the mirroring of oneself in one’s own person,that is, introspection. It becomes reflecting on oneself, on what one did,felt, imagined, thought, on what one is doing, feeling, imagining,thinking. Reflecting, once started, is an activity we perform everymoment, in order to determine our course of action, yet, as a mentalexercise, it can become an aim in itself.


There is however a difference between awareness of the active one of the twobirds and the reflections available from the second bird as witness (see p. 32).

Freudenthal identified several modes of reflection, which he saw as aprocess of shifting one’s standpoint, where the shifting may take place intime, location, or any other mental dimension:


• Reciprocal shifting: shifting from A to B in order to look back at A(looking in a mirror, getting older, making a mental reservation aboutsomething);

• Directed shifting: shifting from A to B while considering C (breaking aprocess down into steps);

• Parallel shifting: shifting A’s environment to B’s.(based on Freudenthal, 1991, p. 105)

See also Freudenthal’s remarks concerning the van Hieles (p. 163).Anne Watson has pointed out that mathematical reflection can be realised

as a rotation if you move up one spatial dimension; psychological reflectionsimilarly involves moving into a different place or dimension, such asDewey’s metaphorical tree.

Reflection: Caleb Gattegno

Gattegno’s writing is rarely clear on first reading, but it is worth the effort toprobe beneath the laconic style which packs ideas into short spaces. Here heconnects reflection, which he sees as a sophisticated activity, with hisfavourite theme of awareness (see p. 61).

… Reflection does not automatically yield its nature so that it can beacknowledged at once as the awareness of awareness because of themovement’s concentration on the substance of the reflection, but it canbe seen for what it is, once the self reaches the dynamics instead of thecontent. Stressing and ignoring, the primitive tools of both plants andanimals, also pervade man’s existence and permit or forbid access towhat is available. Once one is aware of reflection, this could haveyielded the awareness of awareness had the thinker ignored the contentand stressed the dynamics.

(Gattegno, 1987, p. 40)

Note the parallels with Dewey’s views on reflection, though Gattegno isusing reflection to probe even more deeply.

Reflection: Paul Cobb, Kay McClain and Joy Whitenack

… reflective discourse … is characterized by repeated shifts such thatwhat the students and teacher do in action subsequently becomes anexplicit object of discussion. In fact, we might have called itmathematizing discourse because there is a parallel between its struc-ture and psychological accounts of mathematical development in whichactions or processes are transformed into conceptual mathematicalobjects. In the course of the analysis, we also developed the related


construct of collective reflection. This latter notion refers to the joint orcommunal activity of making what was previously done in action anobject of reflection.

(Cobb et al., 1997, p. 258)

Cobb and his colleagues are pointing out similarities between reflection andreification (see p. 167), related to what Piaget called reflective abstraction(see p. 171), and to the MGA spiral (see p. 264).

… The notion of reflective discourse … helps clarify certain aspects ofthe teacher’s role. In our view, one of the primary ways in whichteachers can proactively support students’ mathematical development isto guide and, as necessary, initiate shifts in the discourse such that whatwas previously done in action can become an explicit topic ofconversation.

(ibid., p. 269)

There are similarities with shifts in the structure of attention (p. 60).

… initiating and guiding the development of reflective discourserequires considerable wisdom and judgment on the teacher’s part. Onecan, for example, imagine a scenario in which a teacher persists inattempting to initiate a shift in the discourse when none of the studentsgives a response that involves reflection on prior activity. The very realdanger is, of course, that an intended occasion for reflective discoursewill degenerate into a social guessing game in which students try to inferwhat the teacher wants them to say and do (cf. Bauersfeld, 1980; Voight,1985). In light of this possibility, the teacher’s role in initiating shifts inthe discourse might be thought of as probing to assess whether childrencan participate in the objectification of what they are currently doing.Such a formulation acknowledges the teacher’s proactive role in guidingthe development of reflective discourse while simultaneously stressingboth that such discourse is an interactional accomplishment and thatstudents necessarily have to make an active contribution to itsdevelopment.

A second aspect of the teachers role … is the way in which [theteacher] develop[s] symbolic records of the children’s contributions. Ofcourse one can imagine a scenario in which ways of notating couldthemselves have been a topic for explicit negotiation. For our purpose,the crucial point is not who initiated the development of the notationalschemes, but the fact that the records grew out of the students’ activity ina bottom-up manner … , and that they appeared to play an importantrole in facilitating collective reflection on that prior activity.

(ibid., pp. 269–70)


This is one aspect of what the Do–Talk–Record framework is about (see p. 262).

… What is required is an analytical approach that is fine-grained enoughto account for qualitative differences in individual children’s thinkingeven as they participate in the same collective activities … . Our ratio-nale for positing an indirect linkage between social and psychologicalprocesses is therefore pragmatic and derives from our desire to accountfor … differences in individual children’s activity. As we have noted, thisview implies that participation in an activity such as reflective discourseconstitutes the conditions for the possibility of learning, but it is thestudents who actually do the learning. Participation in reflectivediscourse, therefore, can be seen both to enable and constrain mathe-matical development, but not to determine it.

(ibid., p. 272)

Reflection: Kenneth Zeichner

Ken Zeichner (b. Pennsylvania, 1948–) is an educational researcher who hasmade reflection one of his core interests. Here he draws on three levels ofreflection discerned in organisational psychology: technical, clarifying, andmoral-ethical, based on Dewey’s ideas:

Van Maanen (1977) identifies three levels of reflection, each oneembracing different criteria for choosing among alternative courses ofaction. At the first level of technical rationality … , the dominant concernis with the efficient and effective application of educational knowledgefor the purposes of obtaining ends which are accepted as given. At thislevel, neither the ends nor the institutional contexts of classroom,school, community, and society are treated as problematic.

A second level of reflectivity, … is based upon a conception of practicalaction whereby the problem is one of explicating and clarifying theassumptions and predispositions underlying practical affairs and assessingthe educational consequences toward which an action leads. At this level,every action is seen as linked to particular value commitments, and theactor considers the worth of competing educational ends.

The third level, critical reflection, incorporates moral and ethicalcriteria into the discourse about practical action. At this level the centralquestions ask which educational goals, experiences, and activities leadtoward forms of life which are mediated by concerns for justice, equity,and concrete fulfillment, and whether current arrangements serveimportant human needs and satisfy important human purposes … . Hereboth the teaching (ends and means) and the surrounding contexts areviewed as problematic – that is, as value-governed selections from alarger universe of possibilities.

(Zeichner and Liston, 1987, pp. 24–5)


Reflection: Ingrid Pramling

Pramling develops her view of learning as learning to experience the world(see p. 61), into a list of principles, which include ways in which she usesreflection as a teaching strategy:

• Using reflection as a method in education. To get children to talk andreflect they must become involved in activities (material, situations,play, tasks, etc.) which directly influence them to think about andreflect upon the phenomena about which the teacher wants todevelop their understanding. To get children to talk and reflect inconcrete situations demands that teachers use types of questions thatthey do not normally use, but also to utilize drama, drawing, music,etc. for children to gestalt their understanding.

• Using variation of thought. The teacher must expose the ways inwhich children are thinking and use these as content in education.The teacher must then be aware that children learn from one another,which means that the differences between children are focused oninstead of similarities. Exposing children to variation of thought canbe achieved in many different ways, such as through drawing, drama,play, discussion, etc.

(Pramling, 1994)

Note links with dimensions-of-possible-variation (see p. 56).

Reflection: Grayson Wheatley

In mathematics learning, reflection is characterized by distancing oneselffrom the action of doing mathematics (Sigel, 1981). It is one thing tosolve the problem and it is quite another to take one’s own action as anobject of reflection. In the process of reflection, schemes of schemes areconstructed – the second-order construction. Persons who reflect havegreater control over their thinking and can decide which of several pathsto take, rather than simply being in the action.

It is not enough for students to complete tasks; we must encouragestudents to reflect on their activity. For example, being asked to justify amethod of solution will often promote reflection. This may occur in thesmall-group setting when a learner partner asks, ‘Will that work?’ or itmay occur in the whole-class discussion when the presenter is asked toclarify an explanation. Finally, carefully selected tasks cannot causeperturbation which results in reflection.

[ … ]It is possible that students may be so active that they fail to reflect and

thus do not learn. We can keep students so busy that they rarely havetime to think about what they’re doing, and they may fail to become


aware of the methods and options. In fact, there is an implicit messagethat they are not supposed to think about what they are doing.

(Wheatley, 1992, pp. 535–6)

Wheatley proposes that optimal conditions for learning occur when individ-uals have to defend positions they have taken, and notes that social normshave to be negotiated, including:

1 Group members must assume the obligation of trying to make senseof the explanation;

2 Persons presenting a solution or explanation must present a self-generated solution;

3 … group members recognize the obligation to construct a responseto any challenge to their explanation [with] an explanation whichincorporates a construction of the questioner;

4 The purpose of the dialogue is not to be right but to make sense;5 The purpose of any question raised by a member of the group is to

give meaning to the explanation; it is a sincere and genuine question.(ibid., p. 539)

Compare with scientific debate (see p. 277) and conjecturing atmosphere(see p. 139).

Reflection is one of the ways of awakening the second bird in the image oftwo birds (see p. 32). It is the means whereby the driver of the carriage in theother image can keep the carriage in good condition, look after the horses,and generally be ready to take the owner wherever is required (see p. 33). Itis the means by which it is possible to refresh oneself and to prepare tonotice even more opportunities in the future.


Reflection is much praised but difficult to promote on an ongoing basis.Being stuck is an honourable state.Four phases of problem solving: understanding the problem, making a plan,

carrying out the plan, and looking back.Seven phases of problem solving: getting started, getting involved, mulling,

keeping going, insight, being sceptical, and contemplating.Reflection is a more general response to disturbance, to ‘forked road’

possibilities.Three levels of reflection: technical, clarifying assumptions, and moral-

ethical.

12 Having learned … ?Having learned … ?

Introduction

What is the point of activity arising from tasks? Does it constitute ‘learning’?Presumably learners are expected to have learned something, though someauthors argue that learning cannot be observed: while teaching takes placein time and can be observed, learning takes place over time as a process ofmaturation. Some argue that learning is what we do when we are asleep, asthe brain sorts out the sense impressions of the day, linking some things intopast experiences, and leaving others in relative isolation, which is almosttantamount to forgetting. Activity itself is activity: if there are changes in howactions are perceived, if connections and links are made, then perhapslearning has been facilitated. Since the aim of activity is to enable the growthof understanding, this chapter focuses on knowing, understanding, andobstacles to understanding.

Knowing

Denvir and Brown (1986) showed in their study that some learners improvedperformance on all sorts of tasks, not just the topics they were taught explicitly(see p. 206). This suggests a more general phenomenon, that effective learningof mathematics is more about developing sophistication than about acquiringbatches of isolated skills. Performing some calculations in one domain maycontribute to greater facility in another, ostensibly unrelated domain. Aimingteaching at step-by-step acquisition fails to make use of the more complexmanner in which people develop skills, understanding, and insight.

Inert knowledge: Alfred North Whitehead

Whitehead described the central problem of all education as:

… the problem of keeping knowledge alive, of preventing it frombecoming inert …

(Whitehead, 1932, p. 7)

… Education with inert ideas is not only useless; it is, above all things,harmful … .

(ibid., p. 2)

The notion of inert knowledge (see also p. 35) has been taken up and exam-ined by a multitude of authors seeking both explanation for it and strategiesto overcome it (see Renkl et al., 1996, for a survey; see also rote learning,p. 151).

Knowing: Gilbert Ryle

In his seminal work, Gilbert Ryle (1949) distinguished between knowing-that (factually), knowing-how (to perform acts), and knowing-why (havingstories to account for phenomena and actions). To this can be addedknowing-to (act in the moment as deemed appropriate).

Knowing-why means having some ‘story’ to account for knowing-thator knowing-how, but the story does not have to be valid or true insomeone else’s theories. What we really want is that learners know-to usetechniques they have met and powers they have developed, in newsituations.

These distinctions are not intended to be hard and fast. Some know-howdepends on knowing-that, knowing-to draws upon knowing-how, andknowing-why encompasses knowing-that and knowing-how. But it is oftenthe case that people apparently know-that, yet do not act on that knowl-edge; know-how but fail to recognise an opportunity to employ it; know-why something must be the case but do not use it, or do not use iteffectively.

Learners can often solve routine problems of the type on which they havebeen trained, but as soon as they are given something more general or lessfamiliar, or a task requiring several steps, they are mostly at sea. They do notappear to know-to use what they have learned. For example, Burkhardt(1981) suggested that it is unreasonable to expect learners to use tools formathematical modelling which learners first encountered in the previous twoor three years, and descriptions of the difficulty learners have with multi-stepproblems are legion. It takes time to integrate tools into your own func-tioning, to have them become ‘ert’ (as opposed to inert) in the sense ofWhitehead.

Active, practical knowledge, knowledge that enables people to actcreatively rather than merely react to stimuli with trained or habituatedbehaviour, involves knowing-to act, in the moment. This is what learnersneed in order to engage in problem solving where context is novel and reso-lution non-routine or multi-layered; this is what teachers need in order toprovoke learners into educating their awareness as well as training theirbehaviour (see p. 204). Although teachers believe they are teaching learnersto know actively, their experience suggests otherwise.

Having learned … ? 289

Greeno et al. (1993) use an analogy of motion: motion is not a property ofthe object as it depends upon a frame of reference. Rather, motion is a rela-tion between a frame of reference and an object. So too knowing is not astatic property of a person, but a dynamic and emergent relation betweenperson and situation.

Knowing: Caleb Gattegno

Given his focus on awareness (p. 61), it is natural to find Gattegno basing hisdefinition of knowing on it:

Knowing is the awareness that one is aware of something, and accordingto whether we stress the something or the awareness, we progress in thesubject, or in the education of our awareness. Movements in the educa-tion of our awareness may be short-lived or permanent. When short-lived, they are called flashes of intuition, bright ideas, sudden insights.When permanent, they make possible a familiarity with awareness andprovide a chance to reach awareness of the awareness as a state of beingacknowledged by the self.

(Gattegno, 1987, p. 42)

Unformulated knowing: Brent Davis

Davis takes a strongly enactivist stance towards knowing and under-standing, with a preference for phenomenological approaches to research,and with interests in mathematics education. Here he distinguishes formu-lated and unformulated knowing, based on a distinction of Charles Taylor(1991):

… two sorts of action: formulated and unformulated. The former [Taylor]describes as those thoughts, behaviors, and bits of knowledge that wehave written into the text of our experience – those we are aware of,speak of, and tend to link in narrative and causal chains. Such formu-lated actions, Taylor argues, represent only a small portion of our totalaction, even though they dominate our conscious awarenesses. Thebulk of our moment-to-moment living is a matter of unformulated action– a negotiated movement through an interactive world during which ourknowledge of that world and our way of being in that world are continu-ously enacted. The evidence of such knowledge and understandings isour survival, not our ability to identify, explain (narrate) our actions informal terms.

(Davis, 1996, pp. 44–5)

William James (1899, pp. 33–9) referred to unformulated actions as habitsbelow the level of consciousness. Others have referred to them as automatic


functionings, and even as a form of ‘sleep walking’ in that we have noconscious control over our reactions at those times. The watching bird isasleep (see p. 32). Teaching can be seen in terms of helping learners awakenthat second bird in more situations; working to develop professional practiceis similarly oriented to being more awake to opportunities and to being ableto respond rather than react.

Unformulated knowing is akin to Vergnaud’s theorems-in-action (seep. 63), what Polanyi (1958) called tacit knowing, and what both Vico (1744)and Bachelard (1958) referred to as poetic knowing. For Davis it is in theinterplay between the formulated and the unformulated that learning takesplace.

Knowing: Magdalene Lampert

… Magdalene Lampert posits that there are four types of mathematicalknowledge: intuitive, concrete, computational, and principled concep-tual knowledge (Lampert, 1986). Intuitive knowledge represents anunderstanding that is derived from specific contexts and relates only tothose contexts. Computational knowledge enables one to perform activ-ities with numerical symbols according to previously determined andgeneralizable rules. This is the most common form of mathematicalknowledge presented in schools. Concrete knowledge involvesknowing how to manipulate concrete objects or representations of themto solve a problem. Finally, principled conceptual knowledge representsthe understanding of abstract principles and concepts that govern anddefine mathematical thinking and procedures.

(Merseth, 1993; webref)

Merseth’s article discusses a task very similar to the L’âge du capitaine(Baruk, 1985, see p. 15), which is about errors in mathematics moregenerally.

Transfer

One version of the question of how people come to know-to act in a newsituation is the question of how and when people know to transfer some-thing learned in one context to another context. How are learners to knowto use a technique in an entirely new context? The usual response is thatsomething in the new situation resonates with past experience, bringingthe technique to the surface. But situations one person recognises may notbe the sorts of situations which other people recognise. The term transferwas used initially by people working in the behaviourist tradition. Multipleattempts to locate conditions under which learners could transfer the use ofsome technique or idea to a new context failed to produce any concreteresults (see Detterman and Sternberg, 1993). With the development of


constructivism as a perspective from which to view learning, questionswere raised about transfer as a viable notion. (See also situated cognition,p. 86.) From a situated perspective the issue of transfer becomes the issueof how situatedness broadens and extends: what is it that brings to mindknowing to act in a fresh situation different from that in which the tech-nique has been used before?

Transfer: Lev Vygotsky

Vygotsky considers transitions between using a concept and describing ordefining that concept which resonates with van Hiele phases (see p. 59) andstructure of attention (see p. 60).

The adolescent will form and use a concept quite correctly in a concrete situ-ation but will find it strangely difficult to express that concept in words, andthe verbal definition will, in most cases, be much narrower than might havebeen expected from the way he used the concept. The same discrepancyoccurs also in adult thinking, even at very advanced levels. This confirms theassumption that concepts evolve in ways differing from deliberate consciouselaboration of experience in logical terms. Analysis of reality with the help ofconcepts precedes analysis of the concepts themselves. …

[ … ]The greatest difficulty of all is the application of a concept, finally

grasped and formulated on the abstract level, to new concrete situationsthat must be viewed in these abstract terms – a kind of transfer usuallymastered only toward the end of the adolescent period. The transitionfrom the abstract to concrete proves just as arduous for the youth as theearlier transition from the concrete to the abstract.

(Vygotsky, 1962, pp. 79–80)

Transfer: Overview

Lave (1988) followed up the study by Nunes et al. (1993) with studies of adultnumeracy skills as they left a supermarket (and posed in a supermarket context)and similar tasks posed in their homes. Again a difference in performance wasobserved, both in accuracy and in approach. This led to the notion of situatedcognition: we learn things in a social context, and that context plays an impor-tant role in what is learned as well as in how it is learned (see also Brown,Collins and Duguid, 1989). Transfer then becomes a problem of how situatedcognition expands the range of situatedness. In a new situation, unless there issomething which triggers our attention from past experience, we are unlikely tothink of, or to show evidence of transfer to, the new situation.


Understanding

Some authors focus on knowing and knowledge, with epistemology beingthe study of how we come to know things. But learners of mathematics areexpected to do more than simply know. They are expected to understand,to appreciate, to comprehend. It is often said that we should ‘teach forunderstanding’, in contrast to ‘learning by rote’, which Spencer and others sodetested (see p. 151):

There is a persistent belief in the merits of the goal, but designing schoollearning environments that successfully promote [learning with] under-standing has been difficult.

(Hiebert and Carpenter, 1992, p. 65)

But as we have seen (see p. 289), learning to do, and appreciating why that iswhat you do and why it works, often go hand in hand. Neither alwaysprecedes the other, and a constant diet in one direction stultifies learnersrather than helps them. What then does it mean ‘to understand’, and how canunderstanding be evaluated?

What learners learn: David Wheeler

It is frequently said that we should not teach children to learn mathematicsby rote but that we should teach ‘for understanding’. An obvious imme-diate comment to make is that we cannot imagine any teacher trying toteach for misunderstanding. Of course we must teach for understanding;but how do our pupils achieve it, and how do we know when they have?

It seems as if children may arrive at understanding by different routes.Teaching children to learn by rote does not necessarily prevent themfrom eventually coming to understand, as some of us can testify. Manychildren apparently do not understand mathematics when taught bymethods which require them mainly to imitate and memorise, but we


Knowing-to act in the moment using a particular action or strategy is more sophis-ticated than knowing-that something is true or knowing-how to do somethingwhen asked, and more demanding than knowing-when and knowing-about, oreven knowing-why. All the forms of knowing are involved in understanding,which is a dynamic process and a response to a situation, not a static state.

When training behaviour, there is an issue of whether the learner cantransfer that training to other contexts. When educating awareness, there is anissue of how narrowly situated the awareness is, how contextually bound, andhow that situatedness can be reduced so that learners know-to use techniquesin a broader range of situations.

shall never be able to prove that they would have learned more success-fully if taught another way since they are no longer available for anexperiment. But with both the children who have succeeded and thechildren who have failed, it is apparent that the way mathematics hasbeen taught to them has not entirely controlled the mathematics thatthey have learnt. Some children have understood more than waspresented to them, and some have not understood the little they wereshown.

(Wheeler, 1965, p. 47)

Note the parallel with the findings of Denvir and Brown (see p. 206), and thecontrast with other authors’ views on rote learning (see p. 151).

… It is generally agreed that concepts are not conveyed by instruction,and that some more indirect teaching method is necessary. If conceptsare abstractions, and if they are obtained by fastening onto the commonelements in a number of situations, they can perhaps be taught byputting a variety of situations, all with the same significant feature, infront of the learner. Unfortunately there is no means of knowing inadvance how many examples have to be presented, or any guaranteethat the learner will not choose to concentrate on the irrelevant featureswhen he is faced with a particular situation. A more fundamental objec-tion is that it is the person who has the concept who decides on acommon feature and the ways in which it will be disguised. But whatlooks like a common feature to somebody who is in the know, may notlook like one to somebody who isn’t, and in practice any uncertaintyabout what to abstract is often settled by a procedure not very farremoved from instruction.

(ibid., pp. 47–8)

Note parallels with thinking about generalisation and abstraction (see p. 132and p. 144), and especially same and different (see p. 126). Wheeler’s obser-vations can be seen as a particular instance of the transposition didactique(see p. 83), in that expert awareness is transformed into ‘here, look at this;this is what I am seeing’. See also evaluating understanding (see p. 305).

Understanding: Richard Skemp

To understand something means to assimilate it into an appropriateschema. This explains the subjective nature of understanding and alsomakes clear that this is not usually an all-or-nothing state. We mayachieve a subjective feeling of understanding by assimilation to an inap-propriate schema – the Greeks ‘understood’ thunderstorms by assimi-lating these noisy affairs to the schema of a large and powerful being,Zeus, getting angry and throwing things. In this case, an appropriate


schema involves the idea of an electric spark, so it was not until the eigh-teenth century that any real understanding of thunderstorms waspossible. … Better internal organization of a schema may also improveunderstanding, and clearly there is no stage at which this process iscomplete. One obstacle to the further increase of understanding is thebelief that one already understands fully.

We can also see the deep-rooted conviction … that it matters whetheror not we understand something, is well founded. For this subjectivefeeling that we understand something, open to error though it may be, isin general a sign that we are therefore able to behave appropriately in anew class of situations.

(Skemp, 1971, pp. 46–7)

Relational understanding and instrumental understanding:Richard Skemp

The distinction between instrumental and relational understanding is usuallyattributed to Richard Skemp, who certainly developed and applied it in hiswriting and teaching.

It was brought to my attention some years ago by Steig Mellin-Olsen ofBergen University, that there are in current use two meanings [of reflec-tion]. These he distinguishes by calling them ‘relational understanding’and ‘instrumental understanding’. By the former is meant what I havealways meant by understanding … knowing both what to do and why.Instrumental understanding I would until recently not have regarded asunderstanding at all. It is what I have in the past described as ‘ruleswithout reasons’, without realising that for many pupils and theirteachers the possession of such rule, and ability to use it, was what theymeant by ‘understanding’.

[ … ]If it is accepted that these two categories are both well-filled, by those

pupils and teachers whose goals are respectively relational and instru-mental understanding (by the pupil), two questions arise. First, does thismatter? And second, is one kind better than the other? For years I’vetaken for granted the answers to both these questions: briefly, ‘Yes; rela-tional’. But the existence of a large body of experienced teachers and alarge number of texts belonging to the opposite camp has forced me tothink more about why I hold this view. In the process of changing thatjudgment from an intuitive to a reflective one, I think I have learnt some-thing useful. The two questions are not entirely separate … .

[ … ]Leaving aside for the moment whether one kind is better than the

other, there are two kinds of mathematical mismatches which can occur:


1 Pupils whose goal is to understand instrumentally, taught by ateacher who wants them to understand relationally;

2 The other way about.

The first of these will cause fewer problems short-term to the pupils,though it will be frustrating to the teacher. The pupils just ‘won’t want toknow’ all the careful ground-work he gives in preparation for whateveris to be learned next, nor his careful explanations. All they want is somekind of rule for getting the answer. As soon as this is reached, they latchon to it and ignore the rest.

If the teacher asks a question that does not quite fit the rule, of coursethey will get it wrong. …

(Skemp, 1976, pp. 20–1)

For example: ‘What is the area of the field 20 cm by 15 yds? The reply was‘300 square centimeters’. When asked why not 300 square yards, theyanswered: ‘Because area is always in square centimeters’.

‘Well is the enemy of better’, and if pupils can get the right answers bythe kind of thinking they are used to, they will not take kindly to sugges-tions that they should try for something beyond this.

(ibid., p. 22)

The didactic contract and didactic tension (see p. 79) are another formula-tion of this same idea.

Understanding: Victor Byers and Nicolas Herscovics

Victor Byers and Nicolas Herscovics (1935–1994) extended Skemp’s twoforms of understanding as a result of discussions with teachers. They addedtwo further forms:

• Instrumental understanding is the ability to apply an appropriateremembered rule to the solution of a problem without knowing whythe rule works.

• Relational understanding is the ability to deduce specific rules orprocedures from more general mathematical relationships.

• Intuitive understanding is the ability to solve a problem without prioranalysis of the problem.

• Formal understanding is the ability to connect mathematicalsymbolism and notation with relevant mathematical ideas and tocombine these ideas into chains of reasoning.

(Byers and Herscovics, 1977, p. 26)

Their use of intuitive follows Bruner:


Intuition implies the act of grasping meaning or significance or structureof a problem without explicit reliance on the analytic apparatus of one’scraft … It precedes proof; indeed it is what the techniques of analysisand proof are designed to test and check.

(Bruner, 1965, p. 102, quoted in Byers and Herscovics, 1977, p. 25)

There are interesting comparisons to be made with Ryle, Skemp and others ontypes of knowing (see p. 289), and with Fischbein on intuition (see p. 63).

Modernising their expressions one might wish to do without the ambig-uous notion of ‘the ability’ since understanding is now usually seen as rela-tive to situation, time and place, and not something static possessed by anindividual and available at all times, places, and in all situations. It is alsoworth noting that you can have a direct insight but struggle to express it inany other way; that you can have a sense of how the formal symbols aremanipulated but not appreciate what it is about; that you can know all sortsof connections and links relationally yet not see that the technique applies ina new situation (the difference between knowing-how and knowing-to, seep. 289). Intuitive understanding of a situation is what can be located anddeveloped into explicit and more formal expression (in Dewey’s terms,psychologising the subject matter; in Freudenthal’s terms, locating relevantdidactic phenomena). See also proceptual understanding (p. 166).

Understanding: Edwina Rissland (Michener)

Born in New Jersey, Edwina Rissland (Michener) studied engineeringstudents whom she was teaching at university, trying to find ways of easingtheir difficulties. She produced a categorisation of types of mathematicalobjects which learners need in order to understand a topic and to be able touse techniques effectively, which can be summarised as:

results (theorems, facts); examples (illustrative); and concepts (includingformal and informal definitions and heuristic advice)

(Michener, 1978, p. 362)

Examples are connected with construction methods; results are connectedwith logical deductive reasoning; concepts are connected with pedagogicalordering (ibid., p. 364). Examples themselves have different uses:

start-up examples, reference examples (referred to repeatedly), modelexamples (paradigmatic, generic), counter-examples to conjectures andshowing necessity of hypotheses in theorems

(ibid., pp. 366–8)

Then there are different levels of concepts, such as mega-principles (considerextreme cases, try zero) and counter-principles (watch out for division by zero)


(ibid., pp. 368–9); and different levels of importance of results: key results, tran-sitional results, culminating results (ibid., p. 369). By clarifying the role of exam-ples, principles and results in these terms she found that learners were betterable to make appropriate use of them in making sense of the mathematics.

Understanding: Jeremy Kilpatrick

Jeremy Kilpatrick (b. Iowa, 1935–) was a student of Polya’s, and is a leadingresearcher in mathematics education. Here he and colleagues structure theircomprehensive research-based analysis of the pedagogy of mathematics byseeing understanding as having five strands:

• Conceptual understanding – comprehension and mathematicalconcepts, operations, and relations;

• Procedural fluency – skill in carrying out procedures flexibly, accu-rately, efficiently, and appropriately;

• Strategic competence – ability to formulate, represent, and solvemathematical problems;

• Adaptive reasoning – capacity for logical thought, reflection, explana-tion, and justification;

• Productive disposition – habitual inclination to see mathematics assensible, useful, and worthwhile, coupled with a belief in diligenceand one’s own efficacy.

What does a learner have when they are said to have learned some-thing? Most of us find that we go on learning about things we thought wealready knew about: for example, just what constitutes a number andhow different types of numbers fit together, their properties, and so on,provides an endless field of enquiry.

The products of learning have cognitive, affective and enactivecomponents: you have learned something if you can do things now youcouldn’t do before, perhaps more competently, but perhaps also moreconfidently, moving into the affective domain. There may be a greaterdisposition to think mathematically or to use mathematical thinking inmore contexts, to be sensitised to recognise opportunities for using tech-niques and ways of thinking in new situations. Above all, one expects astronger sense of understanding more, whatever that might mean.

(Kilpatrick et al., 2001, p. 116)

Understanding: Susan Pirie and Tom Kieren

Susan Pirie (b. Kent, 1942–) and Tom Kieren (b. Minnesota, 1940–) togetherdeveloped an onion-layer description of understanding, as a result of tryingto observe it taking place. They noticed that understanding is not a contin-uous process, that sometimes people appear to go backwards, to retreat to


previous ways of thinking, and then suddenly emerge with more sophisti-cated and deeper understanding: hence the layers of onion.

The process of coming to know starts at a level we call ‘primitive doing’.Action at this level may involve physical objects, figures, graphics orsymbols. ‘Primitive’ here does not imply low-level mathematics, but rather astarting place for the growth of any particular mathematical understanding.

The first recursion occurs when the learner begins to form images out ofthis ‘doing’. The effective actions here involve ‘image making’. At the nextlevel these action-tied images are replaced by a form for the images. From themathematical point of view it is this ‘image having’ which frees a person’smathematics from the need to take particular actions as examples. This is afirst level of abstraction; but it is critical to note that it is the learner whomakes this abstraction by recursively building on images based in action. Forunderstanding to grow, these images cannot be imposed from the outside.

(Pirie and Kieren, 1989, p. 8)

There are similarities with reification (see p. 167).

Because knowing has to be effective action, the recursions do not stophere. The images can now be examined for specific or relevant proper-ties. This may involve noticing distinctions, combinations, or connec-tions between images. This level of ‘property noticing’ is the outermostlevel of unselfconscious knowing. (The word ‘outer’ has been carefullychosen to imply that the levels of understanding wrap around eachother, as illustrated in [the] figure, and contain, indeed require, the possi-bility of access to all previous levels. Levels of understanding do notequate with higher or lower levels of mathematics.)


The next level of transcendent recursion entails thinking consciouslyabout the noticed properties, abstracting common qualities anddiscarding the origins of one’s mental action. It is at this level that fullmathematical definitions can occur as one becomes aware of classes ofobjects that one has constructed from the formation of images and theabstraction of their properties.

(ibid., pp. 8–9)

Note parallels with van Hiele levels (see p. 59) and structure of attention (seep. 60).

One is now in a position to observe one’s own thought structures andorganise them consistently. One is aware of being aware, and can seethe consequences of one’s thoughts. It is clear at this point that while thisouter level of understanding is transcendent, in other words fundamen-tally new in some way, it has to be consistent with all previous levels ofknowing.

(ibid., p. 8)

There are similarities with awareness of awareness and the origins of disci-plines (see p. 186 and p. 189).

For fuller understanding one must now be able to answer why theconsequences of thoughts must be true. This calls for an awareness ofassociations and of sequence among one’s previous thoughts, of theirinterdependence. In mathematical terms it might be setting one’sthinking within an axiomatic structure.

All of these levels of recursion are referenced in a direct way toprevious levels. Although new levels transcend or make one free ofactions at an inner ‘level’, in some sense these actions on previous levelsbecome initiating conditions which constrain one’s knowing. At thehighest level of recursion … knowers act as free agents. We call this thelevel of inventing. Now one can choose to initiate a sequence or struc-ture of thought which is a recursion on the previous one in the sense thatit exists as a base, but is freely, yet compatibly, created. …

(ibid., p. 8)

An important feature of their use of this onion image for tracking learners’experiences is that learners frequently ‘fold back’: they behave as if they hadmoved to an inner layer, before then moving out again, perhaps moresecurely. Note the similarities with the spiral model of See–Experience–Master (see p. 263) and with the van Hiele levels or structure of attention (seep. 59).

The National Council of Teachers of Mathematics standards states:


The learning principle: Students must learn mathematics with under-standing, actively building new knowledge from experience and priorknowledge.

(NCTM, webref)

Understanding: Anne Watson

In synthesising research on understanding Watson distinguishes severaldifferent meanings for understanding, each of which plays a role in overallunderstanding:

• Knowing how to perform and use mathematics (instrumental andprocedural);

• Knowing about usefulness in context (contextual);• Relating mathematical concepts (relational);• Knowing about underlying structures (transformable, generalized

and abstract);• Having overcome inherent obstacles.

(Watson, 2002a, p. 153)

She goes on to suggest strategies for promoting these different kinds ofunderstanding.

Experience of understanding

Understanding is more of an experience than a behaviour, so some authorshave tried to elaborate what that experience is like.

Understanding: Janet Duffin and Adrian Simpson

Janet Duffin and Adrian Simpson’s article is unusual in that it tracks their explo-ration of their appreciation of what understanding feels like, stimulated by someclassroom incidents, and in the light of reading of literature generally and onearticle specifically. What follows is a brief extract of some of their conclusions.

We named the three components of understanding building, having,and enacting. By the first component, we mean the formation of theconnections between internal mental structures that we conjectureconstitute the understanding which an individual has ready to be used tosolve problems. With this meaning, it becomes clear that the mecha-nisms for the (formation and destruction) of these connections arealready encapsulated within our theories of responses to natural,conflicting, and alien experiences.

The second component, which we term having, is the state of connec-tions at any particular time. As teachers, it is this totality of connections


(each having the potential to be used by the learner) that we are mostinterested in when we talk about a learner’s understanding. It is impor-tant, at this point, to recognize that the emphasis on ‘connections’ givesthe theory a different way of addressing the issue of whether under-standing is ‘all and nothing’. …

[ … ]By [enacting understanding] we mean the use of the connections

available in the moment to solve a problem or construct a response to aquestion. The breadth of understanding which a learner has may beevidenced by the number of different possible starting points they havefor solving a problem, while the depth of their understanding may beevidenced by the ways in which they can unpack each stage of theirsolution in ever more detail, with reference to more concepts.

[ … ]We can try to determine the internal characteristics by asking

ourselves and other learners the questions:What do I feel when I (am building/have/am enacting) understanding?And we can try to determine the external manifestations by asking

ourselves and other teachers the questions:What do I expect to be able to see in my students when I believe that

they (are building/have/are enacting) their understanding?(Duffin and Simpson, 2000, pp. 420–1)

Obstacles

Gaston Bachelard (b. France, 1884–1962) was a philosopher who focused onthe development of scientific knowledge, informed by a wide range of inter-ests and reading. There are several websites devoted to his memorablequotations. Several of them resonate with the notion of human powers,especially mental imagery (see p. 129).

Man is an imagining being.The words of the world want to make sentences.A word is a bud attempting to become a twig. How can one not dream

while writing? It is the pen which dreams. The blank page gives the rightto dream.

(Bachelard, webref)

Bachelard coined the notion of epistemological obstacles for difficultieslearners experience in getting to grips with certain concepts due to theirintrinsic complexity or sophistication.

It is not a question of considering external obstacles like the complexityor the transient nature of phenomena, nor of implicating weakness ofthe human senses and the human mind; it is in the very act of intimately


knowing that there appear by a sort of functional necessity sluggishnessand troubles … we know against previous knowing.

(Bachelard, 1938, p. 13, quoted in Brousseau, 1997, p. 83)

Bachelard identifies through his study of physics a range of types of obstacle:the obstacle of first experience which will later turn into experience to beconstrued; the obstacle presented by previous general knowledge of thesituation; obstacles produced by the use of particular language; obstaclesarising from inappropriate images and associations; obstacles arising fromfamiliar techniques and actions; obstacles arising from inappropriateanthropomorphisation, assumptions about what is real, and so on, andobstacles arising from factual and quantitative knowledge.

His ideas were developed by Brousseau and Duroux and incorporatedinto the theory of the situation didactique (see p. 79).

Epistemological obstacles and errors: Guy Brousseau

Brousseau links epistemological obstacles with errors learners routinelymake, noting that they often have valuable origins:

… errors and failures do not have the simplified role that we would likethem to play. Errors are not only the effect of ignorance, of uncertainty,of chance, as espoused by empiricist or behaviourist learning theories,but the effect of a previous piece of knowledge which was interestingand successful, but which now is revealed as false or simply unadapted.Errors of this type are not erratic and unexpected, they constitute obsta-cles. As much in the teacher’s functioning as in that of the student, theerror is a component of the meaning of the acquired a piece ofknowledge.

[ … ]We assume, then, that the construction of meaning, as we understand

it, implies a constant interaction between the student and problem-situa-tions, a dialectical interaction (because the subject anticipates anddirects her actions) in which she engages her previous knowings,submits them to revision, modifies them, completes them or rejects themto form new conceptions. The main object of didactique is precisely tostudy the conditions that the situations or the problems put to thestudent must fulfil in order to foster the appearance, the working and therejection of these successive conceptions.

We can deduce from this discontinuous means of acquisition that theinformational character of these situations must itself also change injumps.

(Brousseau, 1997, pp. 82–3)


Note the similarity with conjecturing and with scientific debate (see p. 277),and with Freudenthal’s search for discontinuities (p. 177).

What happens is that [errors] do not completely disappear all at once; theyresist, they persist, then they reappear, and manifest themselves long afterthe subject has rejected the defective model from her conscious cognitivesystem. …

[ … ]The obstacle is of the same nature as knowledge, with objects, relation-

ships, methods of understanding, predictions, with evidence, forgottenconsequences, unexpected ramifications, etc. It will resist being rejectedand, as it must, it will try to adapt itself locally, to modify itself at the leastcost, to optimize itself in a reduced field, following a well-known processof accommodation.

This is why there must be a sufficient flow of new situations which itcannot assimilate, which will destabilize it, make it ineffective, useless,wrong; which necessitate reconsidering it or rejecting it, forgetting it,cutting it up – up until its final manifestation.

(ibid., pp. 84–5)

Brousseau goes on to discuss different origins of obstacles: neurological(due to brain functioning), didactical (due to manner or order of teaching),and epistemological (due to something inherently complex in what is beinglearned) (ibid., pp. 86–87). See also Duroux (1982).

Obstacles: Efraim Fischbein

Fischbein used the notion of obstacles in his study of intuition (see p. 63).

It is highly illuminating to compare the obstacles, difficulties and distor-tions which have appeared in the history of mathematics with those whichemerge during childhood and in the instructional process. Basically, thesame types of conflicts may be identified. Intuitive factors – the quest forpracticality, for behavioral interpretations, for visual, spatially consistentexpressions – have profoundly influenced the historical development ofthe number concept, of the various geometries, of the infinitesimalcalculus, of the concept of infinity, etc. Similar phenomena may bedetected during the instructional process. This supports the hypothesisthat intuitive forms of reasoning are not only a transitory stage in thedevelopment of intelligence. On the contrary, typical intuitive constraintsinfluence our ways of solving and interpreting at every age. Even whendealing with highly abstract concepts, one tends to represent them almostautomatically in a way which would render them intuitively accessible.We tend automatically to resort to behavioral and pictorial representationswhich can confer on abstract concepts the kind of manipulatory features


to which our reasoning is naturally adapted. It has been proved that evenlong after the student has acquired the adequate, highly abstract knowl-edge referring to a certain mathematical notion, the primitive, intuitivemodel on which this notion was originally built may continue to influ-ence, tacitly, its use and interpretation.


Evaluating understanding

Can understanding be assessed or evaluated?

Understanding: David Wheeler

How do we judge successful understanding? There are certain tech-niques – if that is not too grand a word – that we use. Probably thecommonest involves asking the child to perform some task whichappears to require understanding of what he has learnt. If the learningtask was a computational process or algorithm, he will be asked to worksome exercises based on the process. It is not unknown for these to besuccessfully done and yet for very similar exercises to be insoluble amonth later. What has gone wrong? We often say that he cannot reallyhave understood what he was doing the first time. Whether he did ornot, our test of understanding was not adequate as we now want thechild to be able to retain his ability to work the exercises over a period.We have substituted a more demanding requirement.

Possibly we have also met cases of children’s inability to perform a taskthe first time round and an apparently spontaneous emergence of successlater. It is as if germination had gone on unnoticed in the meanwhile. Intro-spection sometimes reveals cases in which we ourselves have suddenlyunderstood – ‘light dawned’, ‘it clicked’ – after a delay in which no externalstimulus seems to have acted. If we are able to learn something when weare not being taught, we can probably imagine the same thing happeningto children. For both of these reasons it can be misleading to ask for a proofof comprehension immediately a piece of instruction has taken place.

Another test of understanding is the application of something learnedto a new problem. This works pretty well as a positive test, but it is aweak indicator if it gives negative results. Failure to make the applicationmay be due to the jump between the new situation and the old being toolarge. …

[ … ]It is not merely perverse to say that we could profitably think less about

‘teaching for (the child’s) understanding’ and more about ‘teaching for theteacher’s understanding’. Only through attending to the second are welikely to make much progress with the first. And the unfortunate truth is


that many of our ways of presenting mathematics in the classroom startfrom the assumption that we (the teachers) already understand and havenothing to learn. Not until we allow that this may not be the case do wefind evidence of our ignorance underneath our noses.

(Wheeler, 1965, pp. 48–50)

Affecting learning: Jean Lave

… if teachers teach in order to affect learning, the only way to discoverwhether they are having effects and if so what those are, is to explorewhether, and if so how, there are changes in the participation of learnerslearning in their various communities of practice. …

… teachers need to know about the powerful identity-changingcommunities of practice of their students, which define the conditions oftheir work. It is a puzzle, however, as to where to find them, and how torecognize them …

(Lave, 1996, pp. 158–9)

Evaluating learning requires agreement on what constitutes learning (see p. 30).For example, if learning includes developing sensitivities to notice moredetail in different situations, then evaluation must include this component; ifit includes changes in the structure of attention and the education of aware-ness, then evaluation must take this into account; if it includes performanceof techniques with facility, then evaluation will take this into account; if itincludes knowing-to use familiar techniques in new contexts, then evalua-tion must offer learners corresponding opportunities; if it includes knowing-about, then evaluation will include opportunities for learners to displayconnections and links, to account for phenomena and to construct relevantnarratives; if it includes constructing mathematical objects which meet speci-fied constraints, then evaluation will include this as a component.

Assessment: Paul Black and Dylan Wiliam

Paul Black is a scientist and science educator who has specialised in assess-ment practices. Dylan Wiliam is a sociologist who researches assessment,with an interest in mathematics education. They have formed the core of ateam of people conducting extensive research into the role of teacher assess-ment (Black and Wiliam, 1998)

In a follow-up booklet (Black et al., 2002) they highlight the importance ofasking suitable questions, extending wait-time (which is easiest if the ques-tions are genuine);

It is the nature, rather than the amount, that is critical when giving pupilsfeedback on both oral and written work. Research experiments have


established that, whilst pupils’ learning can be advanced by feedbackthrough comments, the giving of marks – or grades – has a negativeeffect in that pupils ignore comments when marks are also given (Butler,1988). These results often surprise teachers, but those who have aban-doned the giving of marks find that their experience confirms the find-ings: pupils do engage more productively in improving their work.

(Black et al., 2002, p. 8)

The Turing Test: Alan Turing

Alan Turing (b. London, 1912–1954) is best known for his work decoding theEnigma codes, and the first significant use of a modern computer. Turing’s1950 paper ‘Computing machinery and intelligence’ has become one of themost cited in philosophical literature. In it he proposes first a gender test, butthen shows how the same format provides a test for ‘human cognitive abili-ties’, hence its interest concerning assessment.

I propose … a game which we call the ‘imitation game’. It is played withthree people, a man (A), a woman (B), and an interrogator (C) who maybe of either sex. The interrogator stays in a room apart from the other two.The object of the game for the interrogator is to determine which of theother two is the man and which is the woman. He knows them by labels Xand Y, and at the end of the game he says either ‘X is A and Y is B’ or ‘X isB and Y is A’. The interrogator is allowed to put questions to A and B …

[ … ]In order that tones of voice may not help the interrogator the answers

should be written, or better still, typewritten. The ideal arrangement is tohave a teleprinter communicating between the two rooms. Alternativelythe question and answers can be repeated by an intermediary. Theobject of the game for the third player (B) is to help the interrogator. Thebest strategy for her is probably to give truthful answers. She can addsuch things as ‘I am the woman, don’t listen to him!’ to her answers, butit will avail nothing as the man can make similar remarks.

We now ask the question, ‘What will happen when a machine takesthe part of A in this game?’ Will the interrogator decide wrongly as oftenwhen the game is played like this as he does when the game is playedbetween a man and a woman? These questions replace our original, ‘Canmachines think?’

(Turing, 1950 webref)

The Chinese Room: John Searle

John Searle (b. Colorado, 1932–) is a prolific contemporary philosopher whohas consistently challenged the claims and approaches of ‘the mind–brain as acomputer’ school of thought. He posed a similar problem to Turing’s, called ‘the


Chinese room’. The incoming symbols turn out to be data and questions aboutthe data; the outgoing symbols are answers. Searle supposes that he gets goodenough at using the rules that the answers are indistinguishable from those of aChinese speaker. Despite this, he does not know any Chinese whatsoever.

Suppose I am locked in a room. In this room there are two big bushelbaskets full of Chinese symbols, together with a rule book in Englishfor matching Chinese symbols from one basket with Chinese symbolsfrom the other basket. The rules say things such as ‘Reach into basket1 and take out a squiggle-squiggle sign, and go [and] put that overnext the squoggle-squoggle sign that you take from basket 2. … Nowlet us suppose that the people outside the room send in more Chinesesymbols together with more rules for shuffling and matching thesymbols. But this time they also give me rules for passing backChinese symbols to them. So, there I am in my Chinese room, shufflingthese symbols around; symbols are coming in and I am passing symbolsout according to the rule book. … if I don’t understand Chinese in thatsituation, then neither does any other digital computer solely [by] virtueof being an appropriately programmed computer, …

(Searle, 1987, p. 213)

Searle uses this image to argue that behaviour alone is not sufficient to identifyunderstanding, and that computers will never ‘understand’ text the way humansdo. So what are we doing when testing learners? Understanding requires at leastan affective dimension as well as the cognitive and the behavioural.

Both the Turing test and the Chinese room have similarities with assess-ment, especially in mathematics. As a teacher trying to assess what learnersknow, the problem is very similar: you set them some questions and thenfrom their responses you want to judge the depth of their understanding. Butall you have is their answers. So you grade their answer against a preparedanswer sheet, and where the candidate deviates, you assess the correctnessof their argument. Might they have simply memorised sufficient problem-types so as to recognise the type and then reproduce the technique but withno real understanding of what they were doing or why the technique works?The more sophisticated the questions set, the less well candidates do, somost examiners are content to set questions which are similar to ones previ-ously encountered. Where they differ from familiar ones, fewer candidatesdo well, so teachers train the next year’s candidates in the ‘new type’ of ques-tion. Put another way, how might you test for relational understanding (seep. 295), for educated awareness (see p. 61), for transfer (see p. 291) or flexi-bility in situatedness of learning (see p. 86)?

One of the extra features of assessment is that as soon as an examination istaken, teachers have additional information from which to plan theirteaching for the following year. Within a few years it becomes possible toteach to any test which has a relatively invariant standard format. But


expecting learners to be creative and to tackle novel problems within alimited examination period seems most unreasonable, since so many minorthings can go wrong, obscuring what candidates could do in other circum-stances. Furthermore, you want to enable everyone to show what they cando, and be rewarded, not depress people by making them expose what theycannot (yet) do. And how do you take into account the observation thatlearning is a maturation process which happens over time, since what iscurrently obscure can become clear even when no further attention isdevoted to it?

SOLO taxonomy: John Biggs and Kevin Collis

Inspired by Piaget’s research and insight into how people come to know, butdissatisfied with a stage-theory approach to development, John Biggs andKevin Collis (1982) turned their attention to responses that learners make totasks, in an attempt to distinguish different forms of understanding. Theydeveloped this in the SOLO taxonomy (Structure of the Observed LearningOutcome), and illustrated it with examples drawn from across the schoolcurriculum and over the whole age-span. They distinguished, for example,between prestructural, unistructural, multistructural, relational, andextended abstract responses, as follows:

pre-structural: a reaction which is often a denial of the problem, atautology, or a transduction; bound to specifics, with no apparentconcern for consistency and often no evidence of appreciating theproblem.

uni-structural: an instant reaction to a question stressing one particularfeature which might or might not be relevant; generalises one aspectonly, no evidence of need for consistency, jumps to conclusions onone aspect; tends to close on first idea.

multi-structural: a response which mentions several disparate factors oraspects; aware of consistency but tends to close on a few isolatedfixations on data, and so can reach different conclusions on same dataaccording to what is stressed.

relational: a response which mentions several factors and weaves theminto some sort of related ‘story’ or account; generalises within givenor experienced context using related aspects.

extended abstract: a response which reasons logically using the stressedfeatures as data or as justification; uses deduction and induction,generalises to situations not experienced; attempts to resolve incon-sistencies; satisfied not to close but to leave conclusions open.

(summarised from Biggs and Collis, 1982, pp. 24–5 and pp. 61–93)

For example, a unistructural response to tasks involving the use of algebraicsymbols sees each letter as identified with a particular value (such as the


position in the alphabet). Multistructural responses were satisfied on havingtried a few cases, whereas a relational response appreciates the notion ofgeneralised number, and extended abstract responses showed appreciation ofthe notion of variable beyond that of generalised number (Biggs and Collis,1982, pp. 68–70). The research on which this and other examples in the bookwere based was carried out in the late 1970s, and since then these issues havebeen researched and described in other ways. Nevertheless, the structuralessence of the SOLO levels remain valid and useful for making distinctions.

Apparent or explicit desire for closure was a major feature in the elucida-tion of the taxonomy in mathematics (ibid., p. 67): instant closure on the firstidea (unistructural); being triggered by operations to complete them(multistructural); a sense that questions have single correct answers (rela-tional); or are not always required (extended abstract). Each form ofresponse (rather than stage or level) seems to be specified by absence oflimitations of less sophisticated responses, but the essence of the formscorresponds quite closely with a holistic reaction, recognising distinctions,appreciating relationships, and property use, corresponding to the structureof attention (see p. 60) and van Hiele phases (see p. 59 and p. 163).


Understanding is difficult to capture in words, but has to do with the sensitisa-tion and enriching of awareness, including connections and images, access tolanguage, and extension of powers to make sense of phenomena; and trainingof behaviour in significant procedures and techniques.

Attention is structured, and learning can be seen in terms of shifts in the waywe attend to phenomena, indeed the way we select and identify thephenomena to which we attend.

Learners encounter different kinds of obstacles, inherent in the content,arising from the teaching, and stemming from their own propensities anddispositions.

Testing for understanding is essentially the Turing test. The Chinese roomshows that behaviour alone is not sufficient to determine understanding. Never-theless it is possible to develop sophisticated instruments such as the SOLOtaxonomy for gauging possible or probable understanding from responses totest items. Getting learners to make up their own questions often reveals a gooddeal about what learners are attending to and what they are aware of.

Epilogue

The issue of understanding closes a cycle, for deciding what it means tounderstand is very close to deciding what it means to learn. Teachinginvolves stimulating learners to take initiative, to act and to make use of theirnatural powers in order to make sense of phenomena which attract theirattention. They do this by acting on objects (previously reified) and restruc-turing what they attend to, and what they are sensitised to notice, as well asthrough gaining facility in the use of techniques. Awareness is educated andbehaviour is trained through the harnessing of emotions arising frombecoming aware of a disturbance, of something unexpected. It is the unex-pected which strikes the learners’ attention, and activates the sense-makingapparatus.

The constructs highlighted in this collection are ones which have struck usas editors and practitioners. We hope that at least some of the extracts haveprompted you to want to read more of the authors’ original works, to heartheir voices in a more extended form. Above all, we hope that by usingvarious constructs to make sense of your own experience, you find yourselfstimulated to act in fresh ways, and to find new means of interpreting whatyou notice.

Bibliography BibliographyBibliography

Ahmed, A. 1987. Better Mathematics: A Curriculum Development Study based on theLow Attainers in Mathematics Project (LAMP). HSMO, London.

Ainley, J. 1982. ME234 Using Mathematical Thinking, Unit 9 Mathematical Thinkingin the Primary Curriculum. Open University, Milton Keynes.

Ainley, J. 1987. Telling questions. Mathematics Teaching, 118, pp. 24–6.Anthony, G. 1994. The Role of the Worked Example in Learning Mathematics: SAME

Papers 1994. University of Waikato, Hamilton, pp. 129–43.Armstrong, M. 1980. Closely Observed Children: The Diary of a Primary Classroom.

Writers and Readers, Richmond.Artigue, M. 1993. Didactical engineering as a framework for the conception of

teaching products. In Biehler, R., Scholz, R., Straßer R. and Winkelman, B. (eds),Didactics of Mathematics as a Scientific Discipline. Kluwer, Dordrecht, pp. 27–39.

Atherton, J. webref. Learning and Teaching: Learning from Experience. www.dmu.ac.uk/~jamesa/learning/experien.htm.

Atkinson, R., Derry, S., Renkl, A. and Wortham, D. 2000. Learning from examples:Instructional principles from the worked examples research. Review of Educa-tional Research, 70 (2), pp. 181–214.

Ausubel, D. 1963. The Psychology of Meaningful Verbal Learning. Grune andStratton, New York.

Ausubel, D. 1978. In defense of advance organizers: A reply to the critics. Review ofEducational Research, 48, pp. 251–7.

Ausubel, D. and Robinson, F. 1969. School Learning: An Introduction to EducationalPsychology. Holt, Rhinehart and Winston, New York.

Bachelard, G. 1938 (reprinted 1980). La Formation de l’Esprit Scientifique, J. Vrin, Paris.Bachelard, G. 1958. The Poetics of Space. (Maria, J. trans. 1969). Beacon Press, Boston.Bachelard, G. 1968. The Philosophy of No: A Philosophy of the New Scientific Mind.

(Waterston, G. trans.) Orion Press, New York.Bachelard, G. webref, for example www.creativequotations.com.Balacheff, N. 1986. Cognitive versus situational analysis of problem solving behav-

iours. For the Learning of Mathematics, 6 (3), pp. 10–12.Balacheff, N. 1990. Towards a problématique for research on mathematics teaching.

Journal for Research in Mathematics Education, 21 (4), pp. 258–72.Ball, D. 1992. Magical hopes: Manipulatives and the reform of math education.

American Educator, 16 (2), pp. 14–18, pp. 46–7.Ballard, P. 1928. Teaching the Essentials of Arithmetic. University of London Press,

London.

Banwell, C., Saunders, K. and Tahta, D. 1972 (updated 1986). Starting Points: ForTeaching Mathematics in Middle and Secondary Schools. Tarquin, Diss.

Bartlett, F. 1932. Remembering: A Study in Experimental and Social Psychology.Cambridge University Press, London.

Baruk, S. 1985. L’âge du Capitaine: De l’Erreur en Mathématiques. Éditions du seuil,Paris, p. 23.

Bauersfeld, H. 1980. Hidden dimensions in the so-called reality of a mathematicsclassroom. Educational Studies in Mathematics, 11 (February), pp. 23–41.

Bauersfeld, H. 1988. Interaction, construction and knowledge: Alternativeperspectives for mathematics education. In Grouws, D., Cooney, T. and Jones,D. (eds), Perspectives on Research on Mathematics Education. NCTM andErlbaum, Mahwah, NJ, pp. 27–46.

Bauersfeld, H. 1992. Integrating theories for mathematics education. For the Learningof Mathematics, 12 (2), pp. 19–28.

Bauersfeld, H. 1993. Theoretical perspectives on interaction in the mathematics class-room. In Biehler, R., Scholz, R., Straser, R. and Winkelmann, B. (eds), Didactics ofMathematics as a Scientific Discipline. Dordrecht, Kluwer, pp. 133–46.

Bauersfeld, H. 1995. ‘Language games’ in the mathematics classroom: Their function andtheir effects. In Cobb, P. and Bauersfeld, H. (eds), The Emergence of MathematicalMeaning: Interaction in Classroom Cultures. Erlbaum, Mahwah, NJ, pp. 271–89.

Bell, A. 1986. Diagnostic teaching 2: Developing conflict: Discussion lesson. Mathe-matics Teaching, 116, pp. 26–9.

Bell, A. 1987. Diagnostic teaching 3: Provoking discussion. Mathematics Teaching,118, pp. 21–3.

Bell, A. 1993. Principles for the design of teaching. Educational Studies in Mathe-matics, 24, pp. 5–34.

Benbachir, A. and Zaki, M. 2001. Production d’exemples et de contre-examples enanalyse: étude de cas en première d’université. Educational Studies in Mathematics,47, pp. 273–95.

Bennett, J. 1956–1966. The Dramatic Universe (four volumes). Hodder and Stoughton,London.

Bennett, J. 1993. Elementary Systematics: A Tool for Understanding Wholes. BennettBooks, Santa Fe.

Beth, E. W. and Piaget, J. 1966. Mathematical Epistemology and Psychology. Reidel,Dordrecht.

Biggs J. and Collis K. 1982. Evaluating the Quality of Learning: The SOLO Taxonomy.Academic Press, New York.

Bills, L. 1996. The use of examples in the teaching and learning of mathematics. In Puig,L. and Gutierrez, A. (eds), Proceedings of PME XX, Vol. 2. Valencia, Spain, pp. 81–8.

Bills, L. and Rowland, T. 1999. Examples, generalisation and proof. In Brown, L.(ed.), Making Meaning in Mathematics: A Collection of Extended and RefereedPapers from the British Society for Research into Learning Mathematics, Visions ofMathematics 2, Advances in Mathematics Education 1. QED, York, pp. 103–16.

Black, P. and Wiliam, D. 1998. Inside the Black Box: Raising Standards through Class-room Assessment. School of Education, Kings College, London.

Black, P., Harrison, C., Lee, C., Marshall, B. and Wiliam, D. 2002. Working Inside theBlack Box. Department of Education and Professional Studies, Kings College, London.

Boaler, J. 1997. Experiencing School Mathematics: Teaching Styles, Sex and Setting.Open University Press, Buckingham.

314 Bibliography

Boaler, J. 2000. Exploring situated insights into research and learning. Journal forResearch in Mathematics Education, 31 (1), pp. 113–19.

Boaler, J. (ed.) 2000a. Multiple Perspectives on Mathematics Teaching and Learning,International Perspectives on Mathematics Education. Ablex, London.

Boero, P., Rossella, G. and Mariotti, M. webref. Some Dynamic Mental ProcessesUnderlying Producing and Proving Conjectures. www-didactique.imag.fr/preuve/Resumes/Boero/Boero96.html.

Boero P., Rossella G. and Mariotti M. 1996. Some dynamic mental processes under-lying producing and proving conjectures. In Puig, L. and Gutiérrez, A. (eds),Proceedings of PME XX, Vol. 2, pp. 121–8.

Boole, M. 1901. Indian thought and Western science in the nineteenth century. InBoole, M., (1931), Cobham, E. (ed.), Collected Works, Vol. 3. C. W. Daniel Co.,London, pp. 947–67.

Boulton-Lewis, G. 1998. Children’s strategy use and interpretations of mathematicalrepresentations. Journal of Mathematical Behavior, 17 (2), pp. 219–37.

Bouvier, A. 1987. The right to make mistakes. For the Learning of Mathematics, 7 (3),pp. 17–25.

Brousseau, G. and Otte, M. 1991. The fragility of knowledge. In Bishop, A., Mellin-Olsen, S. and Van Dormolen, J. (eds), Mathematical Knowledge: Its Growththrough Teaching. Kluwer, Dordrecht, pp. 13–36.

Brousseau, G. 1984. The crucial role of the didactical contract in the analysis andconstruction of situations in teaching and learning mathematics. In Steiner, H.(ed.), Theory of Mathematics Education, Paper 54. Institut fur Didaktik derMathematik der Universitat Bielefeld, pp. 110–19.

Brousseau, G. 1986. Fondemonts et méthodes de la didactique des mathématiques,Recherches en Didactique des Mathématiques, 7 (2), p. 33–115.

Brousseau, G. 1997. Theory of didactical situations in mathematics: Didactiques desmathématiques, 1970–1990. (Balacheff, N., Cooper, M., Sutherland, R. and Warfield,V. trans.) Kluwer, Dordrecht.

Brown J. S., Collins, A. and Duguid, P. 1989. Situated cognition and the culture oflearning. Educational Researcher, 18 (1), pp. 32–41.

Brown L. and Coles, A. 1999. Needing to use algebra: A case study. In Zaslavsky, O.(ed.), Proceedings of PME–XXIII, Vol. 2, Haifa, pp. 153–60.

Brown, L. and Coles, A. 2000. Same/different: A ‘natural’ way of learning mathematics.In Nakahara, T. and Koyama, M. (eds.), Proceedings of the 24th Conference of theInternational Group for the Psychology of Mathematics Education, Vol 2. Hiroshima,Japan, pp. 153–60.

Brown, M. 2001. Influences on the teaching of number in England. In Anghileri, J.(ed.), Principles and Practices in Arithmetic Teaching: Innovative Approaches forthe Primary Classroom. Open University Press, Buckingham, pp. 35–48.

Brown, M. and Küchemann, D. 1976. Is it an ‘add’, Miss?: Part 1. Mathematics inSchool, 5 (5), pp. 15–17.



Brown, S. and Walter, M. 1993. Problem Posing: Reflections and Applications.Lawrence Elbaum Associates, Hillsdale, NJ.

Bibliography 315

Bruner, J. 1965 (reprint from 1962). On Knowing: Essays for the Left Hand. Atheneum,New York.

Bruner, J. 1966. Towards a Theory of Instruction. Harvard University Press, Cambridge, MA.Bruner, J. 1986. Actual Minds, Possible Worlds. Harvard University Press, Cambridge, MA.Bruner, J. 1996. The Culture of Education. Harvard University Press, Cambridge, MA.Bruner, J., Goodnow, J. and Austin, G. 1956. A Study of Thinking. Wiley, New York.Bryant, P. 1982. The role of conflict and of agreement between intellectual strategies in

children’s ideas about measurement. British Journal of Psychology, 73, pp. 243–51.Burger, W. and Shaughnessy, J. 1986. Characterizing the Van Hiele levels of development

in geometry. Journal for Research in Mathematics Education, 17 (1), pp. 31–48.Burkhardt, H. 1981. The Real World of Mathematics. Blackie, Glasgow.Bussi, M. webref. Italian Research in Innovation: Towards a New Paradigm?

elib.zib.de/IMU/ICMI/bulletin/45/ItalianResearch.html.Butler, R. 1988. Enhancing and undermining intrinsic motivation: The effects of task-

involving and ego-involving evaluation on interest and performance. BritishJournal of Eduacational Psychology, 58, pp. 1–14.

Byers, V. and Herscovics, N. 1977. Understanding school mathematics. MathematicsTeaching, 81, pp. 24–7.

Caillot, M. 2002. French Didactiques. Canadian Journal of Science, Mathematics andTechnology Education, 2 (3), pp. 397–403.

Calkin, J. 1910. Notes on Education: A Practical Work on Method and School Manage-ment. Mackinlay, Halifax.

Campbell, S. and Dawson, S. 1995. Learning as embodied action. In Sutherland, R. andMason, J. (eds), Exploiting Mental Imagery with Computers in MathematicsEducation. Springer, New York.

Chi, M. and Bassok, M. 1989. Learning from examples via self-explanation. InResnick, L. (ed.), Knowing, Learning and Instruction: Essays in Honor of RobertGlaser. Erlbaum, Mahwah, NJ.

Christiansen, B. and Walther, G. 1986. Task and activity. In Christiansen, B., Howson,G. and Otte, M. (eds), Perspectives in Mathematics Education. Dordrecht, Reidel,pp. 243–307.

Clarke, D. webref. Assessment for Teaching and Learning. http://mcs.open.ac.uk/cme/Clarke_on_tasks1.htm.

Cobb, P. 1991. Reconstructing elementary school mathematics. Focus on LearningProblems in Mathematics, 13 (2), pp. 3–22.

Cobb, P. 1995. Learning and small-group interaction. In Cobb, P. and Bauersfeld, H.(eds), The Emergence of Mathematical Meaning: Interaction in Classroom Cultures.Erlbaum, Mahwah, NJ.

Cobb, P. and Merkel, G. 1989. Thinking strategies: Teaching arithmetic throughproblem solving. In Trafton, P. and Shulte, A. (eds), New Directions for ElementarySchool Mathematics: 1989 Yearbook. National Council of Teachers of Mathematics,Reston, pp. 70–81.

Cobb, P., McClain, K. and Whitenack, J. 1997. Reflective discourse and collec-tive reflection. Journal for Research in Mathematics Education, 28 (3), pp.258–77.

Cobb, P., Wood, T. and Yackel, E. 1991. A constructivist approach to second grademathematics. In Von Glasersfeld, E. (ed.), Radical Constructivism In MathematicsEducation. Reidel, Dordrecht, p. 157–76.

Cobb, P., Yackel, E. and Wood, T. 1992. A constructivist alternative to the representational

316 Bibliography

view of mind in mathematics education. Journal for Research in Mathematics Educa-tion, 23 (1), pp. 2–33.

Colburn, W. 1863. Intellectual Arithmetic: Upon the Inductive Method of Instruction.Houghton, Cambridge.

Cole, M. 1985. The zone of proximal development: Where culture and cognition createeach other. In Wertsch, J. (ed.), Culture, Communication and Cognition: Vygotskianperspectives. Cambridge University Press, Cambridge, pp. 141–61.

Coles, A. and Brown, L. 1999. Meta-commenting: Developing algebraic activityin a ‘community of inquirers’. In Bills, L. (ed.), Proceedings of the BritishSociety for Research into Learning Mathematics MERC. Warwick University,pp. 1–6.

Confrey, J. 1990. What constructivism implies for teaching. In Davis, R., Maher, C. andNoddings, N. (eds), Constructivist Views on the Teaching of and Learning of Math-ematics. NCTM, Reston, pp. 107–22.

Confrey, J. 1991. Steering a course between Vygotsky and Piaget. EducationalResearcher, 20 (2), pp. 29–32.

Confrey, J. 1994. A theory of intellectual development, Part III. For the Learning ofMathematics, 15 (2), pp. 36–45.

Confrey, J. 1994a. A theory of intellectual development, Part I. For the Learning ofMathematics, 14 (3), pp. 2–8.

Confrey, J. 1994b. A theory of intellectual development, Part II. For the Learning ofMathematics, 15 (1), pp. 38–48.

Confrey, J. 1995. How compatible are radical constructivism, sociocultural approaches,and social constructivism? In Steffe, L. and Gale, J. (eds), Constructivism in Educa-tion. Erlbaum, Mahwah, NJ.

Cook, T. 1979. The Curves of Life (reprint of 1914 edition). Dover, New York.Courant, R. 1981. Reminiscences from Hilbert’s Göttingen. Math Intelligencer, 3 (4),

pp. 154–64.Cremin, L. 1961. The Transformation of the School: Progressivism in American

Education, 1876–1957. Knopf, New York.Crowley, M. 1987. The van Hiele model of the development of geometric thought. In

Lindquist, M. and Shulte, A. (eds), Learning and Teaching Geometry K–12: 1987Year Book. NCTM, Reston, pp. 1–16.

Cuoco, A., Goldenberg, P. and Mark, J. 1996. Habits of mind: An organizing principlefor mathematics curricula. Journal of Mathematical Behavior, 15, pp. 375–402.

Da Vinci, L. webref. www-history.mcs.st-andrews.ac.uk/history/Quotations/Leonardo. html.Dahlberg, R. and Housman, D. 1997. Facilitating learning events through example

generation. Educational Studies in Mathematics, 33, pp. 283–99.Davies, H. B. 1978. A seven-year-old’s subtraction technique. Mathematics Teaching,

83, pp. 15–16.Davis, B. 1996. Teaching Mathematics: Toward a Sound Alternative. Garland, London.Davis, B., Summara, D. and Kieren, T. 1996. Cognition, co-emergence, curriculum.

Jouranl for Curriculum Studies, 28, (2), pp. 151–69.Davis, G., Tall, D. and Thomas, M. webref. What is the Object of the Encapsulation of

a Process? www.davidtall.com.Davis, P. and Hersh, R. 1981. The Mathematical Experience. Harvester, Boston and

London.Davis, R. 1966. Discovery in the teaching of mathematics. In Shulman, L. and Keislar,

E. (eds), Learning by Discovery: A Reappraisal. Rand McNally, Chicago.

Bibliography 317

Davis, R. 1984. Learning Mathematics: The Cognitive Science Approach. Croom Helm,London.

De Morgan, A. 1865. A speech of Professor De Morgan, President, at the first meetingof the London Mathematical Society. Proceedings of the London MathematicalSociety, Vol. 1 (1866), pp. 1–9.

De Morgan, A. 1898. Of the Study and Difficulty of Mathematics. Open Court, London.Dearden R. 1967. Instruction and learning by discovery. In Peters, R. (ed.), The

Concept of Education. Routledge and Kegan Paul, London, p. xx.Denvir, B. and Brown, M. 1986. Understanding number concepts in low attaining 7–9

year-olds. Educational Studies in Mathematics, 17 (1), pp. 15–36.Denvir, B. and Brown, M. 1986a. Understanding number concepts in low attaining 7–9

year-olds: Part II. Educational Studies in Mathematics, 17 (2), pp. 143–64.Detterman, D. and Sternberg, R. (eds). 1993. Transfer on trial: Intelligence, Cognition,

and Instruction. Abbex, Norwood, NJ, pp. 99–167.Dewey, J. webref. ‘My Pedagogic Creed’ by John Dewey. www.rjgeib.com/biography/

credo/dewey.html.Dewey, J. 1902. The Child and The Curriculum, (reprinted 1971), The Child and The Curric-

ulum and The School and Society. University of Chicago Press, Chicago, pp. 19–31.Dewey, J. 1933. How We Think: A Restatement of the Relation of Reflective Thinking to

the Educative Process. D.C. Heath and Co., London.Dewey, J. 1938. Experience and Nature. Dover, New York.Dienes, Z. 1960. Building Up Mathematics. Hutchison, London.Dienes, Z. P. 1963. An Experimental Study of Mathematics-learning. Hutchinson

Educational, London.Donaldson, M. 1963. A Study of Children’s Thinking. Tavistock Publications, London.Dörfler, W, 1991. Meaning: Image schemata and protocols: Plenary lecture. In

Furinghetti, F. (ed.), Proceedings of PME XV, Vol. I. Assisi, Italy, pp. 17–33.Dörfler, W. 2002. Formation of mathematical objects as decision making. Mathematical

Thinking and Learning, 4 (4), pp. 337–50.Dubinsky, E. 1991. Reflective abstraction in mathematical thinking. In Tall, D. (ed.),

Advanced Mathematical Thinking. Kluwer, Dordrecht, pp. 95–126.Dubinsky, E. webref. trident.mcs.kent.edu/~edd/.Duffin, J. 1996. Calculators in the Classroom. Manutius Press, Liverpool.Duffin, J. and Simpson, A. 2000. A search for understanding. The Journal of Mathe-

matical Behaviour, 18 (4), pp. 415–28.Duroux, A. webref. 1982. (Warfield, V. trans.) Calculus By Scientific Debate as

an Application of Didactique. www.math.washington.edu/~warfield/articles/Calc&Didactique.html.

Dweck, C. 1999. Self-theories: Their Role in Motivation, Personality and Development.Psychology Press, Philadelphia.

Edwards, D. and Mercer, N. 1987. Common Knowledge: The Development of Under-standing in the Classroom. Routledge, London.

Egan, K. webref. Kieran Egan’s Home Page, www.educ.sfu.ca/kegan/TaSTintro.html.Egan, K. webref a. Getting it Wrong from the Beginning: The Mismatch between

School and Children’s Minds. www.educ.sfu.ca/kegan/Wrong-article.html.Egan, K. 1986. Teaching as Story Telling: An Alternative Approach to Teaching and

Curriculum in the Elementary School. University of Chicago Press, Chicago.Elbaz, F. 1983. Teacher’s Thinking: A Study of Practical Knowledge. Nichols, New York.

318 Bibliography

Enright, M. and Cox, D. webref. Foundations Study Guide: Montessori Education.http://www.objectivistcenter.org/articles/foundations_montessori-education.asp

Erlwanger, S. 1973. Benny’s conception of rules and answers in IPI mathematics.Journal of Children’s Mathematical Behavior, 1 (2), pp. 7–26.

Festinger, L. 1957. A Theory of Cognitive Dissonance. Stanford University Press,Stanford.

Fischbein, E. 1987. Intuition in Science and Mathematics: An Educational Approach.Reidel, Dordecht.

Floyd, A. (ed.) 1981. Developing Mathematical Thinking. Addison Wesley, London.Floyd, A., et al.. 1982. EM235 Developing Mathematical Thinking. Open University,

Milton Keynes.Fox, D. 1983. Personal theories of teaching. Studies in Higher Education, 8 (2),

pp. 151–163.Frankenstein, M. 1989. Relearning Mathematics: A Different R – Radical Math(s). Free

Association, London.Freudenthal, H. 1973. Mathematics as an Educational Task. Reidel, Dordrecht.Freudenthal, H. 1978. Weeding and Sowing: Preface to a Science of Mathematical

Education. Reidel, Dordrecht.Freudenthal, H. 1983. Didactical Phenomenology of Mathematical Structures. Reidel,

Dordrecht.Freudenthal, H. 1991. Revisiting Mathematics Education: China Lectures. Kluwer, Dordrecht.Gagné, R. 1985. The Conditions of Learning (4th edn). Holt, Rinehart and Winston,

New York.Gagné, R., Briggs, L. and Wager, W. 1992. Principles of Instructional Design (4th

edn). HBJ College Publishers, Fort Worth.Gattegno, C. 1963. For the Teaching of Mathematics. Educational Explorers, Reading.Gattegno, C. 1970. What We Owe Children: The Subordination of Teaching to

Learning. Routledge and Kegan Paul, London.Gattegno, C. 1970a. The human element in mathematics. In Mathematical Reflections:

contributions to mathematical thought and teaching (written in memory of A. G.Sillito). Association of Teachers of Mathematics, Cambridge University Press,Cambridge, MA, pp. 131–8.

Gattegno, C. 1974. The Common Sense of Teaching Mathematics. Educational Solu-tions, New York.

Gattegno, C. 1981. Children and mathematics: A new appraisal. Mathematics Teaching,94, pp. 5–7.

Gattegno, C. 1983. On algebra. Mathematics Teaching, 105, pp. 34–5.Gattegno, C. 1987 The Science of Education, Part 1: Theoretical Considerations.

Educational Solutions, New York.Gattegno, C. 1988. The Science of Education, Part 2B: The Awareness of Mathe-

matization. Educational Solutions, New York.Gibson, J. 1977. The theory of affordances. In Shaw, R. E. and Bransford, J. (eds),

Perceiving, Acting, and Knowing. Erlbaum, Mahwah, NJ.Gibson, J. 1979. The Ecological Approach to Visual Perception. Houghton Mifflin, London.Gibson, J. webref. http://www.alamut.com/notebooks/a/affordances.html.Giles, G. 1966. Notes on mathematical activity. In Atkin, B., et al. (eds), The Develop-

ment of Mathematical Ability in Children: The Place of the Problem in this Devel-opment. Association of Teachers of Mathematics, Nelson, pp.8–12.

Bibliography 319

Goldenberg, P., Lewis, P. and O’Keefe, J. 1992. Dynamic representation and thedevelopment of a process understanding of function. In Dubinsky, E. and Harel, G.(eds), The Concept of Function: Aspects of Epistemology and Pedagogy. MAA Mono-graph series, Washington, pp. 235–60.

Goodman, N. 1978. Ways of World Making. Harvester, Boston and London.Gravemeijer, K. 1990. Context problems and realistic mathematics education. In

Gravemeijer, K., Van den Heuvel, M. and Streefland, L. (eds), Contexts, ProductionsTests and Geometry in Realistic Mathematics Education, Technipress, Culemborg.

Gravemeijer, K. 1994. Developing Realistic Mathematics Education. Freudenthal Insti-tute, Utrecht.

Gray, E. and Tall, D. 1994. Duality, ambiguity and flexibility: A proceptual viewof simple arithmetic. Journal for Research in Mathematics Education, 25 (2),pp. 116–140.

Green, D. 1989. School pupils’ understanding of randomness. In Morris, R. (ed.),Studies In Mathematics Education: The Teaching of Statistics, Vol. 7. Unesco, Paris,pp. 27–39.

Greeno, J. 1991. Number sense as situated knowing in a conceptual domain. Journalfor Research in Mathematics Education. 22 (3), pp. 170–218.

Greeno, J. 1994. Gibson’s affordances. Psychological Review, 101 (2), pp. 336–42.Greeno, J., Smith, D. and Moore, J. 1993. Transfer of situated learning. In Detterman, D.

and Sternberg, R. (eds), Transfer on Trial: Intelligence, Cognition and Instruction.Abbex, Norwood, pp. 99–167.

Griffin, P. 1989. Teaching takes place in time, learning takes place over time. Mathe-matics Teaching, 126, pp. 12–13.

Griffin, P. and Gates, P. 1989. Project Mathematics UPDATE: PM753A, B ,C, D, PreparingTo Teach Angle, Equations, Ratio and Probability. Open University, Milton Keynes.

Hall, E. 1981. The Silent Language. Anchor Press, Doubleday, New York.Halmos, P. 1975. The problem of learning to teach. American Mathematical Monthly,

82 (5), pp. 466–76.Halmos, P. 1980. The heart of mathematics. American Mathematical Monthly, 87 (7),

pp. 519–24.Halmos, P. 1985. I Want to be a Mathematician: An Automathography. Springer-

Verlag, New York.Halmos, P. 1994. What is teaching? American Mathematical Monthly, 101 (9),

pp. 848–854.Hamilton, E. and Cairns, H. 1961. The Collected Dialogues of Plato, Bollingen Series

LXXI. Princeton University Press, Princeton.Hamming, R. webref. The Unreasonable Effectiveness of Mathematics. www. lecb.ncifcrf.

gov/~toms/Hamming.unreasonable.html.Hamming, R. 1980. The unreasonable effectiveness of mathematics. American Math-

ematical Monthly, 87 (2), pp. 81–90.Hanson, N. 1958. Patterns of Discovery: An Inquiry into the Conceptual Foundations

of Science. Cambridge University Press.Harel, G. and Kaput, J. 1992. The role of conceptual entities and their symbols in

building advanced mathematical concepts. In Tall, D. (ed.), Advanced Mathemat-ical Thinking. Kluwer, Dordrecht, pp. 82–94.

Harré, R. and van Langenhove, L. (eds). 1999. Positioning Theory: Moral Contexts ofIntentional Action. Blackwell, Oxford.

320 Bibliography

Harré, R. and van Langenhove, L. 1992. Varieties of positioning. Journal for theTheory of Social Behaviour, 20, pp. 393–407.

Hart, K. 1993. Confidence in success. In Hirabayashi, I., Nohda, N., Shigematsu, K.and Lin, F-L. (eds). Psychology of Mathematics Education, PME XVII, Vol. 1, Universityof Tsukuba, Tsukuba, pp. 17–31.

Hart, K, 1981. Ratio and proportion. In Hart, K. (ed.), Children’s Understanding ofMathematics: 11–16. John Murray, London.

Hart, K. 1984. Ratio: Children’s Strategies and Errors. NFER-Nelson, Windsor.Hart, K., Brown, M., Küchemann, D., Kerlsake, D., Ruddock, G. and McCartney, M.

1981. Children’s Understanding of Mathematics. John Murray, London.Healy, L. and Hoyles, C. 2000. Study of proof conceptions in algebra. Journal for

Research in Mathematics Education, 31 (4), pp. 396–428.Hewitt, D. 1994. The Principle of Economy in the Learning and Teaching of Mathe-

matics. Unpublished Ph.D dissertation, Open University, Milton Keynes.Hewitt, D. 1999. Arbitrary and necessary, Part 1: A way of viewing the mathematics

curriculum. For the learning of Mathematics, 19 (3), pp. 2–9.Hiebert, J. and Carpenter, T. 1992. Learning and teaching with understanding. In

Grouws, D. (ed.), Handbook of Research on Mathematics Teaching and Learning.MacMillan, New York, pp. 65–93.

Hiebert, J. and Wearne, D. 1992. Links between teaching and learning place valuewith understanding in first grade. Journal for Research in Mathematics Education,23 pp. 98–122.

Higgins, K. webref. Introduction to Performance Assessment. oregonstate.edu/instruct/ed527/module2/527A2.htm.

Hogben, L. 1938. Clarity is not enough: An address on the needs and difficulties of theaverage pupil. Mathematical Gazette, XXII (249), pp. 105–23.

Holt, J. 1964. How Children Fail. Pitman, London.Holt, J. 1967. How Children Learn. Pitman, New York.Holt, J. 1970. What Do I Do Monday? Dutton, New York.Holton, D. webref. Personal Thoughts on an ICMI Study. Department of Mathematics and

Statistics, University of Otago, New Zealand. www.maths.otago.ac.nz/ICMI_Study.doc.Houssart, J. 2001. Rival classroom discourses and inquiry mathematics: ‘The whisperers’.

For the Learning of Mathematics, 21 (3), pp. 2–8.Hughes, M. 1986. Children and Number: Difficulties in Learning Mathematics.

Blackwell, Oxford.Jackson, P. 1992. Untaught Lessons. Teachers College Press, Columbia University,

New York.James, W. 1899 (reprinted Dover 1961). Talks to Teachers on Psychology and to

Students on Some of Life’s Ideals. Henry Holt, New York.Johnson-Laird, P. and Wason, P. 1977. A theoretical analysis of insight into a

reasoning task. In Johnson-Laird, P. and Wason, P. (eds), Thinking: Readings inCognitive Science. Cambridge University Press, Cambridge, pp. 143–51.

Jong, T. and Ferguson-Hessler, M. 1996. Types and qualities of knowledge. EducationalPsychologist, 31 (2), pp. 105–13.

Jowett, B. 1871. The Dialogues of Plato, Vol. II. Oxford University Press, Oxford.Kahneman, D. and Tversky, A. 1982. Subjective probability: A judgement of representa-

tiveness. In Kahneman, D., Slovic, P. and Tversky, A. (eds), Judgement Under Uncer-tainty: Heuristics and Biases. Cambridge University Press, Cambridge, pp. 32–47.

Kant, I. 1781. (Meiklejohn, J. M. D., trans.) Critique of Pure Reason. MacMillan, London.

Bibliography 321

Kilpatrick, J., Swafford, J. and Findell, B. (eds) 2001. Adding it up: Helping childrenlearn mathematics. Mathematics Learning Study Committee, National AcademyPress, Washington.

King, A. 1993. From sage on the stage to guide on the side. College Teaching, 41 (1),pp. 30–5.

Kitcher, P. 1983. The Nature of Mathematical Knowledge. Oxford University Press, Oxford.Kline, M. 1980. Mathematics: The Loss of Certainty. Oxford University Press, New York.Kolb, D. A. 1984. Experiential Learning: Experience as the Source of Learning and

Development. Prentice-Hall, Englewood Cliffs.Krainer, K. 1993. Powerful tasks: A contribution to a high level of acting and reflecting

in mathematics instruction. Educational Studies in Mathematics, 24, pp. 65–93.Krutetskii, V. 1976. The Psychology of Mathematical Abilities in Schoolchildren.

University of Chicago Press, Chicago.Küchemann, D. 1981. Algebra. In Hart, K., Brown, M., Küchemann, D., Kerlsake, D.,

Ruddock, G. and McCartney, M. (eds), Children’s Understanding of Mathematics.John Murray, London, pp. 102–19.

Laborde, C. 1989. Audacity and reason: French research in mathematics education.For the Learning of Mathematics, 9 (3), pp. 31–6.

Lakatos, I. 1976. Proofs and refutations. In Worral, J. and Zahar, E. (eds), The Logic ofMathematics Discovery. Cambridge University Press, Cambridge.

Lakoff, G. and Johnson, M. 1980. Metaphors We Live By. University of Chicago Press,Chicago.

Lampert, M. 1986. Knowing, doing and teaching multiplication. Cognition andInstruction, 3, pp. 305–42.

Lampert, M. 1990. When the problem is not the question and the solution is not theanswer: Mathematical knowing and teaching. American Educational ResearchJournal, 27 (1), pp. 29–63.

Lampert, M. 2001. Teaching Problems and the Problems of Teaching. Yale UniversityPress, New Haven.

Lave, J. 1988. Cognition in Practice: Mind, Mathematics and Culture in Everyday Life.Cambridge University Press, Cambridge.

Lave, J. 1993. The practice of learning. In Chalkin, S. and Lave, J. (eds), Under-standing Practice: Perspectives on Activity and Context. Cambridge UniversityPress, Cambridge, pp. 3–32.

Lave, J. 1993a. Situated learning in communities of practice. In Resnick, L., Levine, J.and Teasley, T. (eds), Perspectives on socially shared cognition. American Psycho-logical Association, Washington DC, pp. 63–85.

Lave, J. 1996. Teaching as learning. Practice, Mind, Culture and Activity, 3 (3),pp. 149–64.

Lave, J. and Wenger, E. 1991. Situated Learning: Legitimate Peripheral Participation.Cambridge University Press, Cambridge.

Lawler, R. 1981. The progressive construction of mind. Cognitive Science, 5, pp. 1–30.Leapfrogs, 1982. Geometric Images. Association of Teachers of Mathematics, Derby.Legrand, M. 1993. Débate Scientifique en Cour de Mathématiques. Repères IREM, no

10, Topiques Edition.Legrand, M. 1995. Mathématiques, mythe ou réalité, un point de vue éthique sur

l’enseignement scientifique, Repères IREM, 20 and 21. Topiques Editions.Leont’ev, A. 1979. The problem of activity in psychology. In Wertsch, J. (trans., ed.),

The Concept of Activity in Soviet Psychology. Sharpe, New York, pp 37–71.

322 Bibliography

Leont’ev, A. 1981. Psychology and the Language Learning Process. Pergamon, Oxford.Lerman, S. 2000. The social turn in mathematics education. In Boaler, J. (ed.), Multiple

Perspectives on Mathematics Teaching and Learning, International Perspectives onMathematics Education. Ablex, London, pp. 20–44.

Li, S. 1999. Does practice make perfect? For the Learning of Mathematics, 19 (3),pp. 33–5.

Li, Y. and Dù, S. 1987. (Crossley, J. and Lun, A., trans.) Chinese Mathematics: AConcise History. Clarendon Press, Oxford.

Locke, J. 1693. Conduct of Understanding. Quick Edition.Love, E. and Mason, J. 1992. Teaching Mathematics: Action and Awareness. Open

University, Milton Keynes.MacGregor, M. 1991. Making Sense of Algebra: Cognitive Processes Influencing Compre-

hension. Deakin University, Geelong.MacGregor, M. and Stacey, K. 1993. Cognitive models underlying students’ formula-

tion of simple linear equations. Journal for Research in Mathematics Education, 24(3), pp. 217–32.

McIntosh, A. 1977. When will they ever learn? Forum, 19 (3), pp. 92–5.Martino, A. and Maher, C. 1999. Teacher questioning to stimulate justification and

generalization in mathematics: What research practice has taught us. Journal ofMathematical Behavior, 18 (1), pp. 53–78.

Marton, F. 1981. Phenomenography: Describing conceptions of the world around us.Instructional Science, 10, pp. 177–200.

Marton, F. and Booth, S. 1997. Learning and Awareness. Erlbaum, Mahwah, NJ.Marton, F. and Saljo, R. 1976. On qualitative differences in learning. British Journal of

Educational Psychology, 46, pp. 4–11.Mason, J. 1979. Which medium, which message? Visual Education, Feb., pp. 29–33.Mason, J. 1980. When is a symbol symbolic? For the Learning of Mathematics, 1 (2),

pp. 8–12.Mason, J. 1988. ME234 Unit 3. Open University, Milton Keynes.Mason, J. 1989. Mathematical abstraction seen as a delicate shift of attention. For the

Learning of Mathematics, 9 (2), pp. 2–8.Mason, J. 1990. Supporting Primary Mathematics: Space and Shape. Open University,

Milton Keynes.Mason, J. 1992. Doing and construing mathematics in screen space. In Southwell, B.,

Perry, B. and Owens, K. (eds), Space – The First and Final Frontier: Proceedings ofthe 15th Annual Conference of the Mathematics Education Research Group ofAustralasia (MERGA-15), pp. 1–17.

Mason, J. 1996. Expressing generality and roots of algebra. In Berdnarz, N., Kieran, C.and Lee, L. (eds), Approaches to Algebra: Perspectives for Research and Teaching.Kluwer, Dordrecht, pp. 65–86.

Mason J. 1998. Enabling teachers to be real teachers: Necessary levels of awareness andstructure of attention. Journal of Mathematics Teacher Education, 1 (3), pp. 243–67.

Mason, J. 2001. Modelling modelling: Where is the centre of gravity of-for-whenmodelling? In Matos, J., Blum, W., Houston, S. and Carreira, S. (eds), Modelling andMathematics Education: ICTMA 9 Applications in Science and Technology. HorwoodPublishing, Chichester, pp. 39–61.

Mason, J. 2001a. Workshop notes. Johor Baru.Mason, J. 2002. Mathematics Teaching Practice: A Guide for University and College

Lecturers. Horwood Publishing, Chichester.

Bibliography 323

Mason, J. 2002a. Generalisation and algebra: Exploiting children’s powers. InHaggerty, L. (ed.), Aspects of Teaching Secondary Mathematics: Perspectives onPractice. RoutledgeFalmer, London, pp. 105–20.

Mason, J. 2002b. Minding your Qs and Rs: Effective Questioning and Responding inthe Mathematics Classroom. In Haggerty, L. (ed.), Aspects of Teaching SecondaryMathematics: Perspectives on Practice. RoutledgeFalmer, London, pp. 248–58.

Mason, J. 2002c. Researching Your Own Practice: The Discipline of Noticing.RoutledgeFalmer, London.

Mason J., Burton, L. and Stacey, K. 1982. Thinking Mathematically. Addison Wesley,London.

Maturana, H. 1978. Biology of language: The epistemology of reality. In Miller, G. andLenneberg, E. (eds), Psychology and Biology of Language and Thought: Essays inHonor of Eric Lunneberg. Academic Press, New York, pp. 27–63.

Maturana, H. webref. www.hum.auc.dk/~rasand/Artikler/M78BoL.html.Maturana, H. and Varela, F. 1988. The Tree of Knowledge: The Biological Roots of

Human Understanding. Shambala, Boston.Meira, L. 1998. Making sense of instructional devices: The emergence of transparency

in mathematical activity. Journal for Research in Mathematics Education, 29 (2),pp. 121–42.

Mellin-Olsen, S. 1987. The Politics of Mathematics Education. Reidel, Dordrecht.Merseth, K. webref. How Old Is the Shepherd?: An Essay About Mathematics Educa-

tion. lsc-net.terc.edu/do.cfm/paper/8198/show/use_set-papers_pres.Merseth, K. 1993. How old is the shepherd?: An essay about mathematics education.

Phi Delta Kappan, 74 (March 1993), pp. 548–54.Michener, E. 1978. Understanding understanding mathematics. Cognitive Science, 2,

pp. 361–83.Monroe, P. 1909. A Textbook in the History of Education. MacMillan, New York.Montaigne, M. de 1588. (Screech, M. (trans.) 1987). Michel de Montaigne: The

Complete Essays. Penguin, London.Montessori, M. 1912. (George, A. (trans.) reprinted 1964). The Montessori Method.

Schocken Books, New York.Morris, R. (ed.) 1989. Studies in Mathematics Education: The Teaching of Statistics,

Vol. 7. UNESCO, Paris.Movshovits-Hadar, N. 1988. School mathematics theorems – An endless source of

surprise. For the Learning of Mathematics, 8 (3), pp. 34–40.Moyer, P. S. 2001. Are we having fun yet?: How teachers use manipulatives to teach

mathematics. Educational Studies in Mathematics, 47, pp. 175–97.NCTM. webref. http://standards.nctm.org/document/chapter2/learn.htm.Nesher, P. and Winograd, T. 1992. What fifth graders learn when they write their own

math problems. Educational Leadership, 49 (7), pp. 64–7.Nesher, P. 1987. Towards an instructional theory: The role of students’ misconcep-

tions. For the Learning of Mathematics, 7 (3), pp. 33–40.Nesher, P. webref. Towards an Instructional Theory: The Role of Students’ Misconcep-

tions. construct.haifa.ac.il/~nesherp/Public.ht.Newman, J. 1956. The World of Mathematics: A Small Library of the Literature of

Mathematics from A’h-mosé the Scribe to Albert Einstein, presented with commen-taries and notes (four vols.) Simon and Schuster, New York.

Nietz, J. A. 1966. The Evolution of American Secondary School Textbooks. Charles E.Tuttle, Rutland, VT.

324 Bibliography

Northfield, J. and Baird, J. 1992. Learning from the PEEL Experience. Monash Univer-sity Printing Service, Melbourne.

Nunes, T. webref. How Mathematics Teaching Develops Pupils’ Reasoning Systems. Inau-gural Lecture, www.brookes.ac.uk/schools/social/psych/staff/tnicme2000/sld060.htm.

Nunes, T. 1999. Mathematics learning as the socialization of the mind. Mind, Cultureand Activity, 6 (1), pp. 33–52.

Nunes, T., Schliemann, A. and Carraher, D. 1993. Street Mathematics and SchoolMathematics. Cambridge University Press, Cambridge.

Ollerton, M. 2002. Learning and Teaching Mathematics without a Textbook. Associa-tion of Teachers of Mathematics, Derby.

Ollerton, M. and Watson, A. 2001. Inclusive Mathematics: 11–18. Sage, London.Open University. 1978. M101 Mathematics Foundation Course, Block V Unit 1. Open

University, Milton Keynes, p. 31.Orage, A. 1966 (4th edn). On Love: With Some Aphorisms and Other Essays. Samuel

Weiser, New York.Pasteur, L. 1979. Oxford Dictionary of Quotations (3rd edn). Oxford University Press,

Oxford.Papert, S. 1980. Mind Storms. Basic Books, New York.Papert, S. 1993. The Children’s Machine: Rethinking School in the Age of the Computer.

Basic Books, New York. Reproduced at www.stemnet.nf.ca/~elmurphy/emurphy/papert.html.

Papy, G. 1963. Modern Mathematics, Vol. 1. Collier-Macmillan, London.Peirce, C. S. 1902. The essence of mathematics (reprinted 1956). In Newman, J. R.

(ed.), The World of Mathematics. Simon and Schuster, New York, p. 1779.Piaget, J. 1937. La Construction du Réal chez L’enfant. De la Chaux et Niestlé,

Neuchâtel, Switzerland.Piaget, J. and Garcia, R. 1983. Psychogenesis and the History of Science (Feider, H.

trans.) Columbia University Press, New York.Piaget, J. 1970. Genetic Epistemology. Norton, New York.Piaget, J. 1971. Biology and Knowledge: An Essay on the Relations between Organic

Regulations and Cognitive Processes. University of Chicago Press, Chicago.(Originally published in French, 1963.)

Piaget, J. 1972. The Principles of Genetic Epistemology (Mays, W. trans.) Routledgeand Kegan Paul, London.

Piaget, J. 1973. To Understand Is To Invent: The Future of Education. Grossman, NewYork. (Originally published in French, 1948.)

Piaget, J. 1980. Adaptation and Intelligence: Organic Selection and Phenocopy.University of Chicago Press, Chicago. (Originally published in French, 1974.)

Piaget, J. 1985. The Equilibration of Cognitive Structures (Brown, T. and Thampy, K. J.trans.) Harvard University Press, Cambridge, MA. (Original published 1975.)

Pirie, S. and Kieren, T. 1989. A recursive theory of mathematical understanding. Forthe Learning of Mathematics, 9 (3), pp. 7–11.

Pirie, S. and Kieren, T. 1994. Growth in mathematical understanding: How can wecharacterise it and how can we represent it? Educational Studies in Mathematics,26 (2–3) pp. 165–90.

Plowden, B. 1967. The Plowden Report: Children and Their Primary Schools.HMSO, London.

Polanyi, M. 1958. Personal Knowledge. Routledge and Kegan Paul, London.

Bibliography 325

Polya, G. 1954. Induction and Analogy in Mathematics. Princeton University Press,Princeton.

Polya, G. 1957. How to Solve it. Anchor, New York.Polya, G. 1962. Mathematical Discovery: On Understanding, Learning and Teaching

Problem Solving. Combined edition, Wiley, New York.Pramling, I. 1994. Becoming Able: Testing a Phenomenographic Approach to Develop

Children’s Ways of Conceiving the World Around Us. www.ped.gu.se/biorn/phgraph/civil/graphica/oth.su/praml94.html.

Renkl, A., Mandl, H. and Gruber, H. 1996. Inert knowledge: Analyses and remedies.Educational Psychologist, 31 (2), pp. 115–21.

Rig Veda, Samhita 1.164.20.Rosnick, P. and Clement, R. 1980. Learning without understanding: The effect of

tutoring strategies on algebra misconceptions. Journal of Mathematical Behaviour,3 (1), pp. 3–27.

Runesson, U. 1999. The pedagogy of variation: Different ways of handling a mathemat-ical topic. Acta Univertsitatis Gothoburgensis, Göteborg, University of Göteborg.

Runesson, U. 2001. What matters in the mathematics classroom?: Exploring criticaldifferences in the space of learning. In Bergsten, C. (ed.), Proceedings of NORMA01,Kristianstad.

Russell, B. 1926. On Education: Especially in Early Childhood. George Allen andUnwin, London.

Ryle, G. 1949. The Concept of Mind. Hutchinson, London.Schoenfeld, A. 1985. Mathematical Problem Solving. Academic Press, New York.Schoenfeld, A. 1987. Confessions of an accidental theorist. For the Learning of Mathe-

matics, 7 (1), pp. 30–8.Schools Council 1965. Mathematics in Primary Schools, Curriculum Bulletin No. 1.

HMSO, London.Searle, J. 1987. Minds and brains without programs. In Blakemore, C. and Greenfield, S.

(eds), Mindwaves: Thoughts on Intelligence, Identity and Consciousness. Blackwell,Oxford, pp. 209–33.

Selden, A. and Selden, J. webref. Constructivism. www.maa.org/t_and_l/sampler/construct.html.

Sfard, A. 1991. On the dual nature of mathematical conceptions: Reflections onprocesses and objects as different sides of the same coin. Educational Studies inMathematics, 22, pp. 1–36.

Sfard, A. 1992. Operational origins of mathematical notions and the quandary of reifi-cation: The case of function. In Dubinsky, E. and Harel, G. (eds), The Concept ofFunction: Aspects of Epistemology and Pedagogy. Mathematical Association ofAmerica, Washington.

Sfard, A. 1994. The gains and pitfalls of reification: The case of algebra. EducationalStudies in Mathematics, 26, pp. 191–228.

Sfard, A. 1994a. Reification as the birth of metaphor. For the Learning of Mathematics,14 (1), pp. 44–55.

Shiu, C. 1978. The Development of Some Mathematical Concepts in School Children.Unpublished Ph.D thesis, University of Nottingham.

Shuard, H., Walsh, A., Goodwin, J. and Worcester, V. 1991. Calculators, Children andMathematics. Simon and Shuster, London.

Shulman, L. 1987. Knowledge and teaching: Foundations of the new reform. HarvardEducational Review, 57 (1), pp. 1–14.

326 Bibliography

Shulman, L. webref. President of the Foundation: Lee Shulman. www.carnegiefoundation.org/president/biography.htm.

Sierpinska, A. 1987. Humanities students and epistemological obstacles related tolimits. Educational Studies in Mathematics, 18, pp. 371–97.

Sierpinska, A. 1994. Understanding in Mathematics. Falmer Press, London. pp. 78–9.Sigel, I. 1981. Social experience in the development of representational thought:

Distancing theory. In Sigel, I., Brodzinsky, D. M. and Golinkoff, R. M. (eds), Newdirections in Piagetian Theory and Practice. Lawrence Erlbaum, Hillsdale, NJ.

Skemp, R. 1966. The development of mathematical activity in schoolchildren. InBrookes, W. (ed.), The Development of Mathematical Activity in Children: ThePlace of the Problem in this Development. ATM, Nelson, pp. 76–8.

Skemp, R. 1971. The Psychology of Learning Mathematics. Penguin, Harmondsworth.Skemp, R. 1976. Relational understanding and instrumental understanding. Mathe-

matics Teaching, 77, pp. 20–6.Skemp, R. 1979. Intelligence, Learning and Action: A Foundation for Theory and

Practice in Education. Wiley, Chichester.Smith, M. K. 1997. webref. www.infed.org/thinkers/et-mont.htm.Smith, T. 1954. Number: An Account of Work in Number with Children Throughout

the Primary School Stage. Basil Blackwell, Oxford.Snyder, B. 1971. The Hidden Curriculum. Alfred-Knopff, New York.Spencer, H. 1878. Education: Intellectual, Moral and Physical. Williams and Norgate,

London.Spencer, H. 1911. Essays on Education, etc. (Charles W. Eliot, intro.) Dent, London.

(The constituent essays of this collection were first published in the 1850s.)Steffe, L. and Gale, J. 1995. Constructivism in Education. Erlbaum, Mahwah, NJ.Steffe, L., Nesher, P., Cobb, P., Goldin, G. and Greer, B. 1996. Theories of Mathemat-

ical Learning. Erlbaum, Mahwah, NJ.Stein, S. 1987. Gresham’s law: Algorithm drives out thought. For the Learning of

Mathematics, 7 (2), pp. 2–4.Stenmark, J. K. (ed.) 1991. Mathematics Assessment: Myth, Models, Good Questions

and Practical Suggestions. NCTM, Reston, VA.Stigler, J. and Hiebert, J. 1998. Teaching is a cultural activity. American Educator, Winter

1998, pp. 4–11.Stigler J. and Hiebert, J. 1999. The Teaching Gap: Best Ideas from the World’s Teachers

for Improving Education in the Classroom. Free Press, New York.Streefland, L. (ed.) 1991. Realistic Mathematics Education in Primary School. CD-B

Press/Freudenthal Institute, Utrecht, p. 63.Streefland, L. 1991. Fractions in Realistic Mathematics Education. Kluwer, Dordrecht,

p. 50 and p. 63.Tahta, D. 1972. A Boolean Anthology: Selected Writings of Mary Boole on Mathematics

Education. Association of Teachers of Mathematics, Derby.Tahta, D. 1980. About geometry, For the Learning of Mathematics, 1 (1), pp. 2–9.Tahta, D. 1981. Some thoughts arising from the new Nicolet Films, Mathematics

Teaching, 94, pp. 25–9. Reprinted in Beeney, R., Jarvis, M., Tahta, D., Warwick, J.and White, D. (1982) Geometric Images, Leapfrogs, Association of Teachers ofMathematics, Derby, pp. 117-18.

Tahta, D. 1991. Understanding and desire. In Pimm, D. and Love, E. (eds), Teachingand Learning School Mathematics. Hodder and Stoughton, London, pp. 220–46.

Tahta, D. and Brookes, W. 1966. The genesis of mathematical activity. In Brookes,

Bibliography 327

W. (ed.), The Development of Mathematical Activity in Children: The Place of theProblem in this Development. ATM, Nelson, pp. 3–8.

Tall, D. and Vinner, S. 1981. Concept image and concept definition in mathematicswith particular reference to limits and continuity. Educational Studies in Mathe-matics, 12 (2), pp. 151–69. Also www.warwick.ac.uk/staff/David.Tall/pdfs/dot1988e-concept-image-icme.pdf.

Taylor, C. 1991. The dialogical self. In Hiley, D., Bohman, J. and Shusterman, R. (eds),The Interpretive Turn: Philosophy, Science, Culture. Cornell University Press, Ithaca.

Teplow, D. webref. Fresh Approaches: Promoting Teaching Excellence. www.acme_assn.org/almanac/jan97.htm.

Thompson, P. and Thompson, A. 1990. Salient aspects of experience with concretemanipulatives. In Booker, G., Cobb, P. and de Mendicuti, T. (eds), Proceedings ofPME XIV. Oaxtepec, Mexico, pp. 46–52.

Thorndike, E. 1914. Educational Psychology: Briefer Cause. New York: TeachersCollege, Columbia University.

Thorndike, E. 1922. The Psychology of Arithmetic. Macmillan, New York.Thurston, W. 1990. Mathematical education. Notices of the American Mathematical

Society, 37, pp. 844–850. Reprinted in For the Learning of Mathematics, 15 (1)(February 1995), pp. 29–37.

Tizard, B. and Hughes, M. 1984. Young Children Learning: Talking and Thinking atHome and at School. Fontana, London.

Treffers, A. 1986. Three Dimensions: A Model of Goal and Theory Description in Math-ematics Instruction – The Wiskobas Project. Kluwer, Dordrecht.

Turing, A. 1950. Computing machinery and intelligence. Mind: A Quarterly Review ofPhilosophy and Psychology, 59 (236), pp. 433–60.

Turing, A. webref. http://cogprints.ecs.soton.ac.uk/archive/00000499/.Tversky, A. and Kahneman, D. 1982. Judgement under uncertainty: Heuristics and

biases. In Kahneman, D., Slovic, P. and Tversky, A. (eds), Judgement Under Uncer-tainty: Heuristics and Biases. Cambridge University Press, Cambridge, pp. 3–20.

Ulich, R. (ed.) 1999. Three Thousand Years of Educational Wisdom: Selections fromthe Great Documents. Harvard University Press, London.

Van den Brink, J. 1993. Different aspects in designing mathematics education: Threeexamples from the Freudenthal Institute. Educational Studies in Mathematics, 24(1), pp. 35–64.

Van Hiele, P. 1986. Structure and Insight: A Theory of Mathematics Education.Academic Press, Orlando, FL.

Van Hiele-Geldof, D. 1957. The didactiques of geometry in the lowest class ofsecondary school. Reprinted in Fuys, D., Geddes, D. and Tichler, R. (eds) (1984)English Translation of Selected Writings of Dina van Hiele-Geldof and Pierre M.van Hiele. National Science Foundation, Brooklyn College.

Van Lehn, K. 1990. Mind Bugs: The Origins of Procedural Misconceptions. BradfordBook, MIT Press, Cambridge, MA.

Van Maanen, J. (ed.) 1977. Organizational Careers: Some New Perspectives. Wiley, London.Vergnaud, G. 1981. Quelques orientations théoriques et méthodologiques des

recherches françaises en didactique des mathématiques. Actes du Vième Coilloquede PME, Vol. 2. Edition IMAG, Genoble, pp. 7–17.

Vergnaud, G. 1982. A classification of cognitive tasks and operations of thoughtinvolved in addition and subtraction problems. In Carpenter, T., Moser, J. and

328 Bibliography

Romberg, T. (eds), Addition and Subtraction: A Cognitive Perspective. Erlbaum,Mahwah, NJ, pp. 39–59.

Vergnaud, G. 1982a. Cognitive developmental psychology and research in mathe-matics education: Some theoretical and methodological issues. For the Learning ofMathematics, 3 (2), pp. 31–41.

Vergnaud, G. 1983. Multiplicative structures. In Lesh, R. and Landau, M. (eds), Acqui-sition of Mathematics Concepts and Structures. Academic Press, New York, pp.127–74.

Vico, G.-B. 1744. The New Science. (Bergin, T. and Fisch, M. trans. 1961). AnchorBooks, New York.

Voight, J. 1985. Patterns and routines in classroom interaction. Recherches enDidactiques des Mathématiques, 6 (9), pp. 69–118.

Von Glasersfeld, E. 1983. Learning as a constructive activity. PME-NA Proceedings.Montreal, 1983, pp. 41–69.

Von Glasersfeld, E. 1984. An introduction to radical constructivism. In Watzlawick, P.(ed.), The Invented Reality. Norton, London, pp. 17–40.

Von Glasersfeld, E. 1985. Reconstructing the concept of knowledge. Archives dePsychologie, 53, pp. 91–101.

Von Glasersfeld, E. 1991. Abstraction, re-presentation and reflection. In Steffe, L. (ed.),Epistemological Foundations of Mathematical Experience. Springer, New York,pp. 45–67.

Von Glasersfeld, E. 1996. Learning and adaptation in constructivism. In Smith, L.(ed.), Critical Readings on Piaget. Routledge, London, pp. 22–7.

Vygotsky, L. 1962. Thought and Language, MIT Press, Cambridge, MA.Vygotsky, L. 1965. (Hanfmann, E. and Vakar, G. trans.) Thought and Language. MIT

Press, Cambridge, MA.Vygotsky, L. 1978. Mind in Society: the Development of the Higher Psychological

Processes. Harvard University Press, London.Vygotsky, L. 1979. The genesis of higher mental functions. In Wertsch, J. (trans., ed.),

The Concept of Activity in Soviet Psychology. Sharpe, New York, pp. 144–88.Warden, J. 1981. Making space for doing and talking with groups in a primary class-

room. In Floyd, A. (ed.), Developing Mathematical Thinking. Addison Wesley,London, pp. 248–57.

Warfield, V. webref. Calculus By Scientific Debate As An Application Of Didactique.Department of Mathematics, University of Washington. www.math.washington.edu/~warfield/articles/Calc&Didactique.html.

Watson, A. 2001. Low attainers exhibiting higher-order mathematical thinking.Support for Learning, 16 (4), Nov., pp. 179–83.

Watson, A. 2002. Working with students on questioning to promote mathematicalthinking. Mathematics Education Review, 15, pp. 32–42.

Watson, A. 2002a. Teaching for understanding. In Haggerty, L. (ed.), Aspects of TeachingSecondary Mathematics: Perspectives on Practice. RoutledgeFalmer, London.

Watson, A. and Mason, J. 2002. Extending example spaces as a learning/teachingstrategy in mathematics. In Cockburn, A. and Nardi, E. (eds), Proceedings of PME26. University of East Anglia, Vol. 4, p. 377.

Watson A. and Mason, J. 2002a. Student-generated examples in the learning of math-ematics. Canadian Journal of Science, Mathematics and Technology Education, 2(2), pp. 237–49.

Bibliography 329

Watson, A. and Mason, J. (in press). Mathematics as a Constructive Activity: The Roleof Learner-generated Examples. Erlbaum, Mahwah, NJ.

Waywood, A. 1992. Journal writing and learning mathematics. For the Learning ofMathematics, 12 (2), pp. 34–43.

Waywood, A. 1994. Informal writing-to-learn as a dimension of a student profile.Educational Studies in Mathematics, 27, pp. 321–40.

Wertheimer, M. 1961. Productive Thinking (enlarged edition, Wertheimer, E. ed.)Social Science Paperbacks with Tavistock Publications, London.

Wertsch, J. 1991. Voices of the Mind: A Sociocultural Approach to Meditated Action.Harvard University Press, Cambridge, MA.

Wheatley, G. webref. Quick Draw: A Simple Warm-up Exercise Helps Students DevelopMental Imagery of Mathematics. www.learnnc.org/index.nsf/doc/quickdraw.

Wheatley, G. 1992. The role of reflection in mathematics learning. EducationalStudies in Mathematics, 23 (5), pp. 529–41.

Wheeler, D. 1965. Teaching for understanding. Mathematics Teaching, 33, pp. 47–50.Wheeler, D. 1982. Mathematization matters. For the Learning of Mathematics, 3 (1),

pp. 45–7.Whitehead, A. 1911 (reset 1948). An Introduction to Mathematics. Oxford University

Press, London.Whitehead, A. 1932. The Aims of Education and Other Essays. Williams and Norgate,

London.Whiteside, D. 1968. The Mathematical Papers of Isaac Newton, Vol. II, 1667–1670.

Cambridge University Press, Cambridge.Wigner E. webref. The Unreasonable Effectiveness of Mathematics in the Natural

Sciences. www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html.Wigner, E. 1960. The unreasonable effectiveness of mathematics in the natural

sciences. Communications in Pure and Applied Mathematics, 13 (1), pp. 1–14.Winograd, T. and Flores, F. 1986. Understanding Computers and Cognition: A New

Foundation for Design. Ablex, Norwood.Wood, T. 1998. Funneling or focusing? Alternative patterns of communication in

mathematics class. In Steinbring, H., Bartolini-Bussi, M. G. and Sierpinska, A. (eds),Language and Communication in the Mathematics Classroom. National Council ofTeachers of Mathematics, Reston, VA, pp. 167–78.

Wood, D. and Middleton, D. 1975. A study of assisted problem-solving. BritishJournal of Psychology, 66 (2), pp. 181–91.

Wood, D., Bruner, J. and Ross, G. 1976. The role of tutoring in problem solving.Journal of Child Psychology, 17, pp. 89–100.

Yackel, E. 2001. Perspectives on arithmetic from classroom-based research in theUSA. In Anghileri, J. (ed.), Principles and Practices in Arithmetic Teaching. OpenUniversity Press, Buckingham, pp. 15–31.

Zaehner, R. (trans.) 1966. Hindu Scriptures. Dent and Sons, New York.Zammattio, C., Marinoni, A. and Brizio, A. 1980. Leonardo the Scientist. McGraw-Hill,

New York.Zeichner, K. and Liston, D. 1987. Teaching student teachers to reflect. Harvard

Educational Review, 57 (1), pp. 23–48.Zeichner, K. and Liston, D. 1996. Reflective Teaching. Erlbaum, Mahwah, NJ.

330 Bibliography

Index IndexIndex

abilities: learning 119–20abstraction 59, 60, 154; and classification

134–5, 154; education 45; empirical171; learning 59, 60, 154; mathematicseducation 253, 256–7; nature of 133,134; pseudo-empirical 171; reflective171–2

accommodation: learning 149–51actions: APOS theory 172; formulated

290; for learning 153–61;unformulated 290–1

activity: and learning 235–7; mathemati-cal 228–59, 260–79, 280–7

activity theory 84–92; activity levels 85–6;attention 90–1; community of practice95–6; consciousness for two 88–9;internalisation 85; labellingdistinctions 91–2; mediation 85;psychological processes 87, 95; realmof developmental possibilities 89–90;zone of proximal development 85,88, 89, 90

adaptation: learning 149–50affordances: mathematics education

246–7, 259Ainley, Janet: questioning 270–1; using

situations 250–1algebra: and mental imagery 130;

middle years 19; necessity forsurvival 134; use by children 134

America see USAanalysis: and learning 59, 118angles: triangles 22–3, 102, 252APOS theory 172–3apparatus: enactive mode of interaction

3; mathematics teaching 2, 3, 253–9arbitrariness: mathematics 159–60arithmetic: early years 7–9; fractions 18;

horizontal 14–15; middle years 16–17;

Plato’s views 99; street 16–17; vertical14–15

arithmetic teaching: Philip Ballard 50–1;England 50–1; USA 51, see alsomathematics teaching

assessment: understanding 305–10assimilation: in learning 147, 149–51attention: activity theory 90–1; structure

of see structure of attentionAusubel, David: advance organisers

258–9; types of learning 152–3authentic activity: learning motivation

108–10; and zone of proximalrelevance 110

awareness: in articulation 63; educating61–3, 161, 204; in-action 62; andknowing 290; and learning 63;mathematisation 189; nature of 188–9;and reflection 283; of sense 62;structuring 58; and understanding 300

awareness of awareness 189, 290

babies: discernment 125–6Bachelard, Gaston: epistemological

obstacles 302–3BACOMET group: intended curriculum

106–8Balacheff, Nicolas: problématique 83–4,

252Ballard, Philip: arithmetic teaching 50–1Banwell, Colin: children’s learning

abilities 123; invariance amidst change194–5; using situations 249–50

Bartlett, Frederick: memory 31Bauersfeld, Heinrich: activity theory 87;

funnelling 274–5; integrated teaching224; interactionism 76–7

Bell, Alan: activity and learning 235–7;diagnostic teaching 233–5

Biggs, J: SOLO taxonomy 309–10birds, dead: mathematics teaching 250Black, Paul: teacher assessment 306–7Boero, Paolo: task design 241Boole, Mary: children’s learning powers

119; ’fictions’ 175–6; motivation 110–12Booth, Shirley: dimensions-of-variation

56–7; rote learning 152; structuringattention 58

Bouvier, Alain: learning from mistakes208–9

Brookes, Bill: invariance amidst change128–9; mathematics education 122

Brousseau, Guy: didactic contract 79–80;epistemological obstacles and errors303–4; funnelling 275

Brown, Margaret: mathematics learning145–6

Bruner, Jerome: consciousness for two88–9; education 47–8; Enactive–Iconic–Symbolic (EIS) framework 3,260–2, 265; generalisation 137, 269;knowledge 48; learner as active agent148; narrative construction 68–9;personalising curriculum 108;scaffolding 266–8; stances 219; taskdesign 242; three worlds 73–4, 260–2

Burton, Leone: mathematical thinking188

Byers, Victor: understanding 296–7

Campbell, Sen: enactivism 72–3chariot: metaphor for teaching 33–4children: learning abilities 115–24,

183–4; teaching 194; use of algebra134, see also young children

Chinese Room: evaluatingunderstanding 307–9; machineintelligence 307–9

Christiansen, Bent: structure of tasks238–9

Clarke, Doug: task dimensions 243–4class-discussion: as social phenomenon 3classification: and abstraction 134–5,

154; and discrimination 129; ofexperience 123; nature of 126, 195

classroom discussion 277classroom incidents 276Cobb, Paul: realm of developmental

possibilities 89–90; reflection 283–5;teaching dilemma 96–7

cognitive dissonance 69–70cognitive processes: learning 48–9Colburn, Warren: inductive learning 197

Collis, Kevin: SOLO taxonomy 309–10Comenius, John: pansophic movement

37–8; principles of learning 37–8comparison: and learning 118compression: and mathematics 166, 167computers: and learning 40–1concept construction 167–9concept image 200–1concept learning: early years 13–16concepts: definition 292; mathematics

198–202, 297–8; nature of 154–6;teaching 294, see also schemas

conceptual fields 199–200‘the concrete’: mathematics education

254–5condensation: concept construction 169conflict-discussion: learning 234, 237conjecturing 139–42; and intuition 140–1;

mathematics 140, 141–2, 183; andscientific debate 278–9

consciousness for two: activity theory88–9

constraints: mathematics 195constructionism: constructivism 94constructivism 92–8, 148;

constructionism 94; mathematicseducation 83; mathematics learning92; radical 54, 93–4; social 95;teaching dilemma 96–8

constructs: and frameworks 3–4; natureof 2–4

Cook, Theodoret: discrepancies anddiscovery 67–8

counting: early years 10culture: types 74Cuoco, Al: development of learning

powers 181–3curriculum: types 104–8, see also

hidden curriculum

Davis, Bob: paradigm teaching strategy179; reversions 178–9

Davis, Brent: teaching by listening 225–6;unformulated knowing 290–1

Dawson, Sandy: enactivism 72–3de Morgan, Augustus: generalisation

135; mathematics teaching 36deduction: formal 60; informal 59–60Dewey, John: ’the concrete’ 254–5;

discernment 127–8; educationalbeliefs 43–4, 53, 121; psychologisingsubject matter 45–6, 230–3; purposeof education 45; reflection 280–1;reification 127–8

332 Index

didactic contract 79–83, 84; learner–teacher interaction 81, 82–3, see alsosituation didactique

didactic phenomenology: mathematics202–3

didactic situations: theory 79–84didactic tension: learner–teacher

interaction 82–3didactique des mathématiques 80–1Dienes, Zoltan: generalisation 136–7;

multiple embodiment 253–4; playand learning 247–8

differences: discernment of 124–9;recognition by children 126

dimensions-of-possible-variation 57–8,see also range-of-permissible-change

dimensions-of-variation 56–8discernment: learning 124–9discovery: and education 117; from

discrepancies 67–8; and teaching 224discovery learning 248–9discrepancies: and discovery 67–8discrimination: and classification 129;

and learning 118distribution patterns: middle years 19–20disturbance: and education 118; and

learning 149; and motivation oflearners 101, 103–4, 161; andnarrative 68–9

Dörfler, Willi: image schemata 201–2;objects 169–70

Dubinsky, Ed: APOS theory 172–3;reflective abstraction 171–2

Duffin, Janet: understanding 301–2Dweck, Carol: learner self-esteem 112–13

early years: concept learning 13–16;counting 10; making sense 10–12;mathematics education 7–16; shaperecognition 9; subtraction 12–13;written arithmetic 7–9

Easter bunnies: mathematics teaching250–1

education: and abstraction 45; aware-ness 61–3, 161, 204; as discipline38–9; and discovery 117; anddisturbance 118; learner-centred 49;nature of 43–4, 52–3; philosophy of120–1; psychological basis 44;purpose 33, 45; rhythms 46–7;science of 61–2; sociological basis 44,see also learning; teaching

educational psychology 39–40

Edwards, Derek: positioning 218–19;teaching dilemma 97–8

Egan, Kieran: educational beliefs 116;imagination 131

enactive mode of interaction: apparatus3; frameworks 3

enactivism 70–3England: arithmetic teaching 50–1epistemological obstacles see obstaclesepistemology: genetic see constructivism;

mathematics education 83equilibration: and learning 148–50errors: and learning 303–4evaluation: machine intelligence 307–9;

understanding 305–10examples: learning from 173–4existence: mathematical entities 65experience: classification of 123; and

learning 118experiencing 61experimentation: and learning 118;

mathematics 182explicitation: learning 163–4

feedback: and learning 237Festinger, Leon: cognitive dissonance

69–70‘fictions’: and teaching 175field of experience: task design 241Fischbein, Efraim: conjecturing and

intuition 140–1; intuition and learning63–7; obstacles 304–5

formalisation: learning 124Fox, Dennis: metaphors for teaching 42–3frameworks: and constructs 3–4; Do–

Talk–Record (DTR) 262–3, 265–6;enactive mode of interaction 3;Enactive–Iconic–Symbolic (EIS) 3,260–2, 265, 266; Manipulating–Getting-a-sense-of–Articulating(MGA) spiral 264–5, 266; See–Experience–Master (SEM) 263–4, 266

free orientation: learning 164Freudenthal, Hans: concepts 199;

context-rich mathematics 251; didacticphenomenology 202–3; mathematicsteaching 49–50, 143; mathematisation189–90; nature of teaching 143;prospective and retrospective learning164, 165; reflection 282–3; teaching asguiding 223–4

fundamental triad 74–5funnelling: teaching 274–5

Index 333

Gagné, Robert: conditions for learning48–9

Gattegno, Caleb: abstraction 134;awareness 188–9; discernment 126;educating awareness 61–2; knowing290; learners’ powers 121–2, 183–4;learning in children 183–4; learningfrom mistakes 209–10; mentalimagery 130; nature of mathematics185–6; reflection 283; science ofeducation 61–2; teaching andlearning 226–7

generalisation: language 132; andlearning 138–9; mathematics 132–7,139, 185; mathematics teaching 229;nature of 133; and specialisation 137–9;and symbolism 136–7

genetic epistemology see constructivismgeometry: non-Euclidean 103–4Germany: mathematics teaching 220Gibson, James: affordance 246Giles, Geoff: activity and education 248Goldenberg, Paul: development of

learning powers 181–3Gravemeijer, Koeno: use of apparatus

254Gray, Eddie: procepts 166–7Greeno, James: affordance 246–7growth theory: teaching 42–3guided orientation: learning 163

Halmos, Paul: mathematics as problemsolving 187–8; specialising 137;starting with a question 228–30

Harré, Rom: positioning 218Hart, Kath: use of apparatus 255–6Herscovics, Nicolas: understanding

296–7Hewitt, Dave: arbitrary knowledge

159–61; subordinating 179–80hidden curriculum: motivation of

learners 104–5Hiebert, James: teaching as cultural

activity 40–1; teaching triads 220–1higher psychological processes: activity

theory 87, 95Hilbert, David: generalisation 133Hogben, Lancelot: mathematics

teaching 100; motivation 100Holt, John: funnelling 274, 275; learner-

centred education 49; learners’ habits272–3; motivation 111; teaching asguiding 223

Holton, Derek: scientific debate 278–9

horse and chariot: metaphor forteaching 33–4

human psyche: as two birds 32–3

imagination 129–31; and learning 73–4inert knowledge 52, 288–9information: learning 163integration: learning 164intellect: development 121interactionism 76–7interiorisation: concept construction

168–9; process construction 172intuition: and conjecturing 140–1; and

learning 45–6, 63–7; and mathematics63–7

invariance amidst change 127, 128–9,193–5

invention: mathematics 183irrational numbers 103

Jackson, Philip: classroom incidents 276Japan: mathematics teaching 220Jesuits: principles of learning 37

Kant, Immanuel: intuition and learning45–6

Kieren, Tom: understanding 298–301Kilpatrick, Jeremy: understanding 298knowing: and awareness 290; and

learning 289–90; nature of 53; types of289–90, 293, 301; unformulated 290–1

knowledge: arbitrary 159–61; computa-tional 291; concrete 291; inert 52,288–9; intuitive 291; mathematical 291;nature of 93; pedagogical content 41;principled conceptual 291; transfer291–3; types 41–2, 291

Kolb, David: learning cycle 161–2Krainer, Konrad: task design 239–41Krutetskii, Vadim: generalisation 139;

learning abilities 119–20

labelling distinctions: activity theory 91–2Laborde, Colette: situation didactique

80–1Lampert, Magdalene: complexity of

teaching 217–18; conjecturing 140;knowing 291; learning throughproblems 225

language: and generalisation 132;nature of 134; and teaching 93

later years: probability 28; problemsolving 28; proof 26–8; reasoning 25–9;statistics 25–6, 28

334 Index

Lave, Jean: community of practice 95–6;evaluating learning 306

learners: habits 272–3; mistakes 208–10;natural powers 115–42; purposes 275;self-esteem 112–13; types ofexperience 102–3

learner–teacher interaction 81–3learning: abilities 115–24; and abstraction

59, 60, 154; accommodation 149–51; asaction 70–5, 143–80; active 146, 147–8,157–9, 161; and activity 235–7; adap-tation 149–50; advance organisers258–9; assimilation 147, 149–51;awareness 61–3; biological basis148–51; cognitive processes 48–9;by comparison 118; computers 40–1;conditions for 30–78; conflict-discussion 234, 237; cultural practice78; development of powers 181–4;and discernment 124–9; discoverymethods 159; disturbance 149;equilibration 148–50; errors 208–10,303–4; evaluation 306; from examples173–4; experimentation 118;explicitation 163–4; exploratory 147;feedback 237; formalisation 124, 147;free orientation 164; generalisation138–9; guided orientation 163;imagination 73–4; inductive 197;information 163; integration 164;intuition 45–6, 63–7; knowing 289–90;laws of 40; making distinctions 55–8;material 73–4; maturation 263;meaningful 151–3; mechanism of 206–8;misconceptions 212–13; from mistakes208–10, 303–4; motivation 146–7;nature of 52–78, 206–8, 288; optimalconditions 287; phases 147, 161–5; play247–8; positioning 218–19; practice175–7; principles 37–8, 75–6, 146–7;prospective 164, 165; purposes 68;recitation 37; repetition 237; responseto disturbance 67–70; retrospective 164,165; schemas 154–9; skills 174–80;social interaction 75–8; stages of 162–3,165; stressing and ignoring 126–8;structuring attention 58–61;subordination 179–80; symbolic 73–4;and teaching 226–7; through problems225, see also education; mathematicseducation; mathematics learning;mathematics teaching; teaching

learning abilities: children 115–124,183–184

learning actions: mathematics 144–6Legrand, Marc: scientific debate 277–8Leonardo da Vinci: importance of

theory 30–1Leont’ev, Aleksej: activity theory 85–6Lewin, Kurt: learning cycle 161–2Li, Shiqi: practice and learning 176–7listening: and teaching 225–6, 227Locke, John: education as discipline 38–9Love, Eric: mathematics learning actions

144–5lower primary school see early yearslower psychological processes: activity

theory 87Loyola, Ignatius: recitation as learning 37

machine intelligence: Chinese Room307–9; Turing Test 307

Maher, Carolyn: questioning 273making sense: early years 10–12manipulatives see apparatusMark, June: development of learning

powers 181–3Martino, Amy: questioning 273Marton, Ference: dimensions-of-variation

56–7; phenomenography 55–6; rotelearning 152; structuring attention 58

Mason, John: children’s learningabilities 124; conjecturing atmosphere141–2; dimensions-of-possible-variation 57–8; generalisation 137;mathematical modelling 191–2;mathematical thinking 188; mathe-matics learning actions 144–5;meaning of tasks 241–2; sources ofquestions 272; specialising 138;teacher–learner interactions 221–2;teaching children 194

material world: and learning 73–4mathematical entities: existence 65mathematical knowledge: types 291mathematical objects: classification

297–8mathematics: activity 228–59, 260–79,

280–7; arbitrariness 159–60;compression 166, 167; concepts198–202, 297–8; conjecturing 140,141–2, 183; constraints 195; description182; development of learning powers181–4; experimentation 182; generali-sation 132–7, 139, 185; initiating activity228–59; intuition 63–7; invarianceamidst change 127, 128–9, 193–5;invention 183; learning 176–7, 301;

Index 335

learning actions 144–6; manipulationof ideas 151; meanings 195–6; mentalimagery 130, 131; modelling 190–3;nature of 153–4, 185–6; paradigmteaching strategy 179; pattern finding182; pragmaticism 185; problemsolving 186–8; procedural thinking167; procepts 166–7; teacher–learnerinteraction 220–2; thinking 184–93;topics 198–206; transformation 165;understanding 293–310; undoing 193;variability principle 253; visualisation183, see also mathematics education;mathematics learning; mathematicsteaching

mathematics education 122; abstraction253, 256–7; affordances 246–7, 259;applicability 99–100; ‘the concrete’254–5; constructivism 83; context 251;didactic phenomenology 202–3; earlyyears 7–16; epistemology 83;phenomena 2–4; problématique 83–4;using situations 245–52, see alsoeducation; mathematics; mathematicslearning; mathematics teaching

mathematics learning: adaptivereasoning 298; conceptualunderstanding 298; constructivism 92;motivation 99–114; proceduralfluency 298; productive disposition298; products 298; reflection 280–7;schemas 156; strategic competence298; tasks 238–45; and teaching 80–1;and understanding 293–302, see alsolearning; mathematics; mathematicseducation; mathematics teaching

mathematics teaching: apparatus 2, 3,253–9; generalisation 229; andlearning 80–1; open-field exercises177–8; purpose 39; reversions 178–9;set theory 108–9; structure-of-a-topic203–5; techniques 196–7, see alsoarithmetic teaching; mathematics;mathematics education; mathematicslearning; teaching

mathematisation: awareness 189;horizontal 189–90; modelling 192;vertical 189–90

Maturana, Humberto: enactivism 70–2meanings: in mathematics 195–6memory: functioning 31mental imagery 129–31; and algebra

130; and mathematics 130, 131

Mercer, Neil: positioning 218–19;teaching dilemma 97–8

middle years: algebra 19; arithmetic16–17; distribution patterns 19–20;fractions 18; proof 23–5; relativereasoning 20–3; sorting 18; use ofsymbols 19

misconceptions: learning 212–13mistakes: learning from 208–10, 303–4modelling: mathematical 190–3;

mathematisation 192Montessori, Maria: educational

philosophy 120–1mothers: tutoring children 268–9motivation: authentic activity 108–10;

goals 100–1; hidden curriculum104–5; intended curriculum 106–8;learner self-esteem 112–14; learning146–7; mathematics learning 99–114;surprise 101–2, 103–4, 161; teacherdesire 110–12

Movshovits-Hadar, Nitsa: motivation bysurprise 101–2

Moyer-Packenham, Patricia: use ofapparatus 257–8

narrative: and disturbance 68–9Nesher, Pearla: learning from mistakes

212–13numbers: irrational 103Nunes, Terezinha: children’s learning

abilities 115; learning and culture 78

objects: APOS theory 172–3; nature of169–70

observation: and theory 31–2obstacles: learning errors 303–4, 310;

origins 304; to understanding 302–5,310; types 303

onion-layer model: understanding298–300

Open University: mathematicalmodelling 190–1; structure-of-a-topic203–5

ordering 195orientation: learning 163, 164Otte, Michel: didactic contract 79–80

pansophic movement: teachingmethods 37–8

Papert, Seymour: constructionism 94Papy, Georges: authentic activity

108–10

336 Index

paradigmatic examples: teaching 174pattern finding: mathematics 182pedagogic strategy: talking-in-pairs 3Peirce, Charles Saunders: pragmaticism

185perceptual variability principle 253Pestalozzi, Johann: arithmetic teaching

51; education 39phenomena: mathematics education 2–4phenomenography 55–6phenomenology: didactic 202–3philosophy: of education 120–1Piaget, Jean: use of apparatus 257;

biological basis of learning 148–50;constructivism 92, 148; inaccuracy oftranslations 54; learning actions 153–4;learning stages 162–3, 165

Pirie, Susan: understanding 298–301Plato: on arithmetic 99; on education 34–5play: classification 247–8; and learning

247–8Polya, George: active learning principles

146–7; conjecturing 140; mathematicsas problem solving 186–7

positioning: and learning 218–19practice: learning skills 174–80pragmaticism: mathematics 185Pramling, Ingrid: experiencing 61;

reflection 286probability: later years 28problem solving: later years 28;

mathematics 186–8; phases 280, 287;teaching 229

problématique: mathematics education83–4

procedural thinking: mathematics 167procepts: mathematics 166–7processes: APOS theory 172proof: later years 26–8; middle years

23–5psyche, human see human psychepsychologising subject matter 45–6,

49–50, 230–3psychology: Soviet 119Pythagoras’ theorem: surprisingness 102

questioning: and teaching 269–76

range-of-change-change 57–8, see alsodimensions-of-possible-variation

realm of developmental possibilities89–90

reasoning: adaptive 298; later years 25–9recitation: learning 37

reflection: and awareness 283; aseducational method 286; levels of 285,287; mathematics learning 280–7;modes of 282–3

reflective abstraction 171–2reflective discourse 283–5reification 127–8; concept construction

167–9relative reasoning: middle years 20–3repetition: and learning 118, 237research-uncertainty principle: see also

structuring attentionresonance 128rich contexts see mathematics educa-

tion: using situationsRissland (Michener), Edwina: under-

standing 297–8Ross, Gail: scaffolding 266–8; task

design 242rote learning: dangers 36, 151–3;

meaning of 152–3; value of 152rule-teaching 196ruptures: irrational numbers 103; and

motivation of learners 103–4; non-Euclidean geometry 103–4

Russell, Bertrand: children’s learningabilities 116–17

Russia see USSRRyle, Gilbert: knowing 289–90

Saunders, Ken: children’s learningabilities 123; invariance amidst change194–5; using situations 249–50

scaffolding: properties 267–8; teaching266–76

schemas: disadvantages 156–7; andlearning 154–9; mathematics learning156, see also concepts

science: of education 61–2scientific debate 277–9scientific enquiry: nature of 127Searle, John: evaluating understanding

307–9secondary school see later yearsself-esteem: motivation of learners 112–14set theory: mathematics teaching 108–9Sfard, Anna: reification 167–9shape recognition: early years 9shaping theory: teaching 42–3Shulman, Lee: types of knowledge 41–2Sierpinska, Anna: exemplary examples

173–4; fundamental triad 74–5similarities: discernment of 124–9;

recognition by children 126

Index 337

Simpson, Adrian: understanding 301–2situation didactique 79, 80–1Skemp, Richard: abstraction 134–5;

accommodation in learning 150–1;assimilation in learning 150–1;children’s learning abilities 123;classification of experience 123;learning schemas 154–9; motivation100–1; reflection 281–2;understanding 294–6

skills: learning through practice 174–80Smith, Thyra: discernment in babies 125–6Snyder, Benson: hidden curriculum

104–5SOLO taxonomy: evaluating under-

standing 309–10sorting: middle years 18specialisation: and generalisation 137–9Spencer, Herbert: children’s learning

abilities 115, 116; discovery learning248–9; on education 35–6, 39; learneras active agent 147–8; rote learning152; teaching techniques 196

Stacey, Kaye: mathematical thinking 188stances see positioningstatistics: later years 25–6, 28; probability

28Stein, Sherman: open-field exercises

177–8, 244–5Stigler, James: teaching as cultural

activity 40–1; teaching triads 220–1structure of attention 60Structure of the Observed Learning

Outcome see SOLO taxonomystructuring attention: learning 58–61structuring awareness see structuring

attentionsubtraction: early years 12–13;

understanding 200, 205–6surprise: and motivation 101–2, 103–4survival: necessity of algebra 134symbolism: and generalisation 136–7;

and learning 73–4symbols: use of 19

Tahta, Dick: children’s learning abilities123; invariance amidst change 128–9,194–5; mathematics education 122;meaning of tasks 241–2; usingsituations 249–50

talking-in-pairs: pedagogic strategy 3Tall, David: concept image 200–1;

procepts 166–7task design 239–41, 242

tasks: classification 242–5; mathematicslearning 238–45; meaning of 241–2;open-field 177–8, 244–5

teacher desire: and motivation 110–12teachers: assessment 306–7; role of 215,

217–27teacher–learner interaction:

mathematics 220–2; wait-time 2, 3teaching: child-sensitive 49; children

194; complexity 217–18; concepts294; cultural activity 40–1; diagnostic233–5; discovery 224; ‘fictions’ 175;funnelling 274–5; growth theory 42–3;guiding 223–4; as horse and chariot33–4; integrated 224; and language93; and learning 226–7; by listening225–7; and mathematics learning 80–1;metaphors for 33–4, 42–3; nature of41–2, 143, 217–18; paradigmaticexamples 174; problem solving 229;questioning 269–76; scaffolding266–76; shaping theory 42–3; startingwith a question 228–30; story telling131; and training 175; transfer 42–3,291–3; travelling theory 42–3, see alsoeducation; learning; mathematicseducation; mathematics learning;mathematics teaching

teaching dilemma: constructivism 96–8teaching triads 220–2tertiary education see later yearstheory: importance of 30–2thinking: levels of 59–60; mathematical

184–93; nature of 117–18; reflective280–7; verbal 282

Thorndike, Edward: educationalpsychology 39–40; laws of learning 40

three worlds: learning as action 73–4,260–2

Thurston, William: compression 166Topaze effect 275topic structure: mathematics 202–6training: nature of 93; and teaching 175transfer: and teaching 42–3, 291–3transformation: mathematics 165travelling theory: teaching 42–3Treffers, Adrian: mathematisation

189–90triangles: angles of 22–3, 102, 252Turing, Alan: evaluating understanding

307; machine intelligence 307Turing Test: machine intelligence 307tutoring: young children 268–9two birds: as human psyche 32–3

338 Index

understanding: and awareness 300;conceptual 298; evaluation 305–10;formal 296; instrumental 295–6;intuitive 296–7; levels of 298–302;mathematics 293–310; nature of 294–5, 310; obstacles 302–5; onion-layermodel 298–300; relational 295–6

undoing: mathematics 193Upanishads: and human psyche 32–4;

and teaching 33–4USA: arithmetic teaching 51;

mathematics teaching 220USSR: psychology 119

van den Brink, Jan: teaching principles237–8

van Hiele, Dina and Pierre: learningphases 163–4, 165; levels of thinking59–60; structuring attention 59–60

van Lehn, Kurt: learning from mistakes210–11

Varela, Francisco: enactivism 70–2variability principle: mathematics 253Vergnaud, Gerard: conceptual fields

199–200Vico, Gianbattista: nature of knowing 53Vinner, Shlomo: concept image 200–1visualisation 59; mathematics 183von Glasersfeld, Ernst: adaptation in

learning 150; radical constructivism54, 93–4

Vygotsky, Lev: activity theory 84–5, 87–8;attention 90–1; generalisation 136;labelling distinctions 91–2; transfer 292;zone of proximal development 85, 88

wait-time: teacher–learner interaction 2, 3Walther, G.: structure of tasks 238–9

Warden, Janette: classroom discussion 277Watson, Anne: dimensions-of-possible-

variation 57–8; questioning 271–2;understanding 301

Wenger, Etienne: principles of learning75–6

Wertheimer, Max: nature of thinking117–18

Wheatley, Grayson: abstraction 256–7;imagery 131; reflection 286–7

Wheeler, David: evaluatingunderstanding 305–6; mathematicseducation 122; mathematisation 192;understanding mathematics 293–4

Whitehead, Alfred North: discovery andeducation 117; educational rhythms46–7; generalisation 132–3; inertknowledge 288–9; nature of education52–3

Wigner, Eugene: invariance 194Wiliam, Dylan: teacher assessment

306–7Wood, David: scaffolding 266–9; task

design 242

young children: learning abilities 183–4;mathematical concepts 125; tutoring268–9, see also children

Zeichner, Kenneth: reflection 285Zhoubi Suanjing: learning and generalising

138–9zone of proximal development:

activity theory 85, 88, 89, 90;realm of developmental possibilities90

zone of proximal relevance: authenticactivity 110

Index 339